Topic 6: Spatial Interpolation

第六讲

空 间 插 值

Chapter Outline

13.1 Introduction

Box 13.1 A Survey of Spatial Interpolation among GIS Packages

13.2 Elements of Spatial Interpolation

13.2.1 Control Points

13.2.2 Type of Spatial Interpolation

13.3 Global Methods

13.3.1 Trend Surface Analysis

13.3.2 Regression Models

13.4 Local Method

Box 13.2 A Worked Example of Trend Surface Analysis

13.4.1 Thiessen Polygons

13.4.2 Density Estimation

Box 13.3 A Worked Example of Kernel Estimation

13.4.3 Inverse Distance Weighted Interpolated

Box 13.4 A Worked Example of Using the Inverse Distance Weighted Method for Estimation

13.4.4 Thin-plate Splines

Box 13.5 Radial Basis Functions

Box 13.6 A Worked Example of Thin-plate Splines with Tension

13.4.5 Kriging

13.4.5.1 Ordinary Kriging

Box 13.7 A Worked Example of Using Ordinary Kriging for Estimation

13.4.5.2 Universal Kriging

Box 13.8 A Worked Example of Using Universal Kriging for Estimation

13.4.5.3 Other Kriging Methods

13.5 Comparison of Spatial Interpolation Methods

Box 13.9 Spatial Interpolation using ArcGIS

Applications: Spatial Interpolation

Task 1: Use Trend Surface Analysis for Global Interpolation

Task 2: Use Kernel Density Estimation for Local Interpolation

Task 3: Use IDW for Local Interpolation

Task 4: Compare Two Splines Methods

Task 5: Use Ordinary Kriging for Local Interpolation

Task 6: Use Universal Kriging for Local Interpolation

Task 7: Use Cokriging for Local Interpolation

What is Spatial Interpolation?

Spatial interpolation is the process of using points with known values to estimate values at other points. These points with known values are called known points, control points, sampled points, or observations.

In GIS applications, spatial interpolation is typically applied to a grid with estimates made for all cells. Spatial interpolation is therefore a means of converting point data to surface data so that the surface data can be used with other surfaces for analysis and modeling.

A map of 105 weather stations in Idaho and their 30-year average annual precipitation values

Spatial interpolation

Classification of Spatial Interpolation

1. Global vs. Local

2. Exact vs. Inexact

3. Deterministic vs. Stochastic

Global vs. Local

A global interpolation method uses every known point available to estimate an unknown value.

A local interpolation method, on the other hand, uses a sample of known points to estimate an unknown value.

(a) (b) (c)

Three search methods for sample points: (a) find the closest points to the point to be estimated, (b) find points within a radius, and (c) find points within each of the four quadrants.

Exact vs. Inexact

Exact interpolation predicts a value at the point location that is the same as its known value. In other words, exact interpolation generates a surface that passes through the control points. In contrast, inexact interpolation or approximate interpolation predicts a value at the point location that differs from its known value.

Deterministic vs. Stochastic

A deterministic interpolation method provides no assessment of errors with predicted values. A stochastic interpolation method, on the other hand, offers assessment of prediction errors with estimated variances.

Global Local

Deterministic Stochastic Deterministic Stochastic

Trend surface (inexact)*

Regression (inexact)

Thiessen (exact)

Density estimation (inexact)

Inverse distance weighted (exact)

Splines (exact)

Kriging (exact)

*Given some required assumptions, trend surface analysis can be treated as a special case of regression analysis and thus a stochastic method.

A classification of spatial interpolation methods

An isoline map of a third-order trend surface created from 105 points with annual precipitation values

The diagram shows points, Delaunay triangulation in thinner lines, and Thiessen polygons in thicker lines.

The simple density estimation method is used to compute the number of deer sightings per hectare from the point data.

The kernel estimation method is used to compute the number of sightings per hectare from the point data.

An annual precipitation surface map created by the inverse distance squared method

An isohyet map created by the inverse distance squared method

An isohyet map created by the regularized splines method

An isohyet map created by the splines with tension method

                           

  

Five mathematical models for fitting semivariograms: Gaussian, linear, spherical, circular, and exponential

A semivariogram constructed from annual precipitation values at 105 weather stations in Idaho. The linear model provides the trend line.

An isohyet map created by ordinary kriging with the linear model

The map shows the standard deviation of the annual precipitation surface created by ordinary kriging with the linear model.

An isohyet map created by universal kriging with the linear drift

The map shows the standard deviation of the annual precipitation surface created by universal kriging with the linear drift.

The map shows the difference between surfaces generated from the regularized splines method and the inverse distance squared method. A local operation, in which one surface grid was subtracted from the other, created the map.

The map shows the difference between surfaces generated from the ordinary kriging with linear model method and the regularized splines method.

Comparison of Local Methods

Comparison of local methods is usually based on statistical measures, although some studies have also suggested the importance of the visual quality of generated surfaces such as preservation of distinct spatial pattern and visual pleasantness and faithfulness.

Cross validation and validation are two common statistical techniques for comparison.

Applications

廖顺宝等 . 气温数据栅格化的方法及其比较 . 资源科学 , 2003,25(6):83 ～ 88.

Liao Shun-bao et al. Comparison on Methods for Rasterization of Air Temperature Data . Resources Science, 2003,25(6):83 ～ 88.

龙 亮等 . 基于 GIS的精准农业信息流分析方法研究 . 资源科学 , 2003,25(6):89 ～ 95.

Long Liang et al. Information Flow and Interpolation Methods of GIS for Percision Agriculture. Resources Science, 2003,25(6):89 ～ 95

Conservation Reserve Program (CRP), Farm Service Agency (FSA) of the U.S. Department of Agriculture

http://www.fsa.usda.gov/dafp/cepd/crp.htm

California LESA model

http://www.consrv.ca.gov/DLRP/qh_lesa.htm/

WEPP (Water Erosion Prediction Project)

http://topsoil.nserl.purdue.edu/nserlweb/weppmain/wepp.html

SWAT

http://waterhome.tamu.edu/NRCSdata/SWAT_SSURGO

Better Assessment Science Integrating point and Nonpoint Sources (BASINS) system

http://www.epa.gov/waterscience/BASINS/

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