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LSGI 3421: GIS Applications Lecture 1: Introduction to GIS ApplicationsLSGI 3244: Spatial Analysis Lecture 7: Spatial Interpolation & Surface Analysis

LSGI 3244: Spatial Analysis

Dr. Bo Wu [email protected]

Department of Land Surveying & Geo-InformaticsThe Hong Kong Polytechnic University

Lecture 7: Spatial Interpolation and

Surface Analysis

LSGI 3244: Spatial Analysis Lecture 7: Spatial Interpolation & Surface Analysis

1. Learning Outcomes2. 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

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• 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

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• 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

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• 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

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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

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• Linear vs Non-Linear Interpolation


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• Exact vs Inexact– Exact interpolation

• Predicts a value at the point location that is the same as its known value

• Generate a surface that passes through the control points

– Inexact interpolation• Predicts a value at the point location that differs from its known

value (approximate interpolation)

• Deterministic vs Stochastic– Deterministic interpolation

• Provides no assessment of errors with predicted values

– Stochastic interpolation• Offers assessment of prediction errors with estimated variances

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Typical Spatial Interpolation Methods

• Regression• Splines• Thiessen polygons (Voronoi polygons) • Trend surface• Inverse distance weighted (IDW)• Kriging

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• Trend Surface Models– Inexact interpolation method– Using polynomial equation for

approximation, e.g.• 1st order & one dimension trend

face: z = b0 + b1x + e• 1st order & two dimension trend

face: z = b0 + b1x + b2y + e

– Least-squares method is used

Trend Surface

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Inverse Distance Weighted (IDW)

• 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

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• 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:

Kriging

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• Simple Kriging– assumes a known constant trend: μ(x) = 0.

• Ordinary Kriging– assumes an unknown constant trend: μ(x) = μ.

• Universal Kriging– assumes a general linear trend model

• Other Kriging Methods

Types of Kriging

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Characteristics of Kriging• Use the semivariogram, in calculating estimates of the

surface at the grid nodes• Assume spatial variation may consist of 3 components

– A structural component, representing a trend– A spatially correlated component, representing the

variation of the regionalized variable– A random error term

• Can assess the quality of prediction with estimation prediction errors (stochastic)

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3 Components of a Spatial Variable

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The structural component (e.g., a linear trend)

The random noise component (non-fitted)The spatially correlated component


Semivariogram in Kriging

• Semivariogram– Measure the spatial dependence or spatial autocorrelation of a

group of points

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ϒ(h) is the semi-variance between known point xi and xj, separated by the distance h; and z is the attribute value

or



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a

( ) ( ) ⎥⎦⎤

⎢⎣⎡ −+= 223 3

10 ahahcchγ

( ) 10 cch +=γ

( ) 00 =γ

h=0 • Sphere model:

0 < h < a

h >= a

h=0

• Exponent model:

( ) ⎥⎦

⎤⎢⎣

⎡ −+= )

3exp(10 a

hcchγ h > 0

( ) 00 =γ h=0

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Ordinary Kriging

• 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

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• 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)


Ordinary Kriging

• The weight W can be determined by solving a set of simultaneous equations:

• The variance can be estimated by:

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DWC =⋅


Numeric Example of Kriging

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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 and their distance to point 0.


Spatial Dependence Analysis

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Parameters:

C0 = 0, a = 10, C1 = 10

( ) ⎥⎦

⎤⎢⎣

⎡ −+= )

3exp(10 a

hcchγ


Kriging Matrices

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

λ


Kriging Matrices

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• First, the distance matrix

• Then, the exponent model will be used to calculate the semivariogram matrix

C(h) = 10 e –0.3|h|( ) ⎥⎦

⎤⎢⎣

⎡ −+= )

3exp(10 a

hcchγ

C0 = 0, a = 10, C1 = 10

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Results

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Kriging weights:

Estimated value for point 0:

λ

How can the interpolation variance be estimated?


Universal Kriging

• 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

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Ordinary Kriging vs Universal Kriging

• Ordinary Kriging– Without drift

• Universal Kriging– With drift

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• Slope– The slope at a point is the angle

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

• Aspect– The angle formed by moving clockwise

from north to the direction of maximum slope

Analysis of Surface

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Y SLOPE

NORTH

ASPECT


Surface Analysis - Slope

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• b = (z3 + 2z6 + z9 - z1 - 2z4 - z7) / 8D • c = (z1 + 2z2 + z3 - z7 - 2z8 - z9) / 8D

– 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.


Surface Analysis - Slope

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• Slope does not indicate the direction of the calculated slope.

• Using ArcMap to create a slope surface, click on Spatial Analyst/Surface Analysis/Slope.

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Surface Analysis - Aspect

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• Aspect is a neighborhood function which creates a grid of aspect or direction of maximum slope of the cells of the input grid.

• tan (aspect) = b/c− b denotes slope in the x

direction − c denotes slope in the y

direction• Aspect values are in degrees

with 0° for the North direction.• To create a aspect surface,

click on Spatial Analyst/ Surface Analysis/Aspect.


Surface Analysis - Viewshed

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• Viewshed is a global function which creates a grid of visible and non-visible surface from an observation point.

• To create a viewshed grid, click on Spatial Analyst/ Surface Analysis/ Viewshed.


Surface Analysis - Hillshade

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• Hillshade is a neighborhood function which creates a grid of surface brightness for a given position of a light source.

• Hillshade values can be used to enhance the legend of themes.

• To create a hillshade surface, click on Spatial Analyst/Surface Analysis/ Hillshade.

Modeling incoming solar radiation (a–f) representing

morning to evening


• Create contours creates a line feature dataset in which the lines connect points of equal cell value.

• Using ArcMap to create contours, click on Spatial Analyst/Surface Analysis/Contour.

Surface Analysis - Contour

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Night launch of “Opportunity” in July 2003

An Example in NASA’s Mars Exploration Rover Mission (MER 2003)

Launch of “Spirit” in June 2003

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Seven Months Later ……

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Spirit:Launched: June 10, 2003Landed on Mars: Jan. 4, 2004

Opportunity:Launched: July 7, 2003Landed on Mars: Jan. 25, 2004

Mars Rover in MER 2003 Mission

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Images Taken by the MER Rovers

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First Panorama taken by the Opportunity Rover

Opportunity looks back at itsempty lander as it begins toexplore the Meridiani Planum.


Martian Surface

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Images Taken by the MER Rovers

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Navcam panorama at Duck Bay taken by the Opportunity Rover

Pancam panorama at Duck Bay taken by the Opportunity Rover

Rover tracks


Spatial Data Handling Cycle

Step I: Data Acquisition

on Martian Surface

Step IV: Science and Engineering

Activity Planning

Step II: Pre-processing at NASA

Rover Localization

Map Product Generation

Product Distributionvia WebGIS

Step III: Science Data Processing

(e.g., Spatial Data Processing Map Product Distribution at OSU)

Earth

via Mars Odyssey via MGS

DTE

Mars

DSN

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Investigate the Victoria Crater

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Sol 953

Mapping at Duck Bay

Sol 1204

Sol 1210

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Measured 3D Points

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DEM Interpolated from the 3D Points

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3D Surface of Duck Bay

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Contour Map from DEM

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Slope Map from DEM

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3D Terrain and Slope Maps of Duck Bay

LegendSlope (degree)

0 - 5

5 - 10

10 - 15

15 - 20

20 - 25

25 - 80

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Opportunity Traverse for Descending into the Victoria Crater


Long-time Water and Winds Signatures

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The Road To Endeavour

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• Further readings– Kriging.com (http://www.kriging.com/)– Caroline Lafleur, 1998, MATLAB Kriging Toolbox.

(http://globec.whoi.edu/software/kriging/V3/english.html)– Li, R., B. Wu, et al., 2008. Characterization of Traverse Slippage of Spirit Rover

on Husband Hill at Gusev Crater, Mars. Journal of Geophysical Research-Planets, 113, E12S35, doi:10.1029/2008JE003097.

• Summarization of the main ideas presented in this lecture:

• Questions?

Review

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