Uses one of five basis functions to process each measured sample value, thus creating an exact interpolation surface

Interpolation

true

#### Authorship

:
View
Is authorship not correct? Feed back

#### Classification(s)

Method-focused categoriesData-perspectiveGeoinformation analysis

#### Detailed Description

English {{currentDetailLanguage}} English

## Summary

Uses one of five basis functions to process each measured sample value, thus creating an exact interpolation surface.

## Usage

• The smooth search neighborhood is only available for the Inverse multiquadric function.

• For all methods except the Inverse multiquadric function, the higher the parameter value, the smoother the surface. The opposite is true for the Inverse multiquadric function.

## Code Sample

Interpolate point features onto a rectangular raster.

import arcpy
arcpy.env.workspace = "C:/gapyexamples/data"
"ca_ozone_pts", "OZONE", "outRBF", "C:/gapyexamples/output/rbfout", "2000",
arcpy.SearchNeighborhoodStandard(300000, 300000, 0, 15, 10, "ONE_SECTOR"),
"THIN_PLATE_SPLINE", "")


Interpolate point features onto a rectangular raster.

# Name: RadialBasisFunctions_Example_02.py
# Description: RBF methods are a series of exact interpolation techniques;
#              that is, the surface must go through each measured sample value.
# Requirements: Geostatistical Analyst Extension

# Import system modules
import arcpy

# Set environment settings
arcpy.env.workspace = "C:/gapyexamples/data"

# Set local variables
inPointFeatures = "ca_ozone_pts.shp"
zField = "OZONE"
outLayer = "outRBF"
outRaster = "C:/gapyexamples/output/rbfout"
cellSize = 2000.0
rbf = "THIN_PLATE_SPLINE"
smallscaleParam = ""

# Set variables for search neighborhood
majSemiaxis = 300000
minSemiaxis = 300000
angle = 0
maxNeighbors = 15
minNeighbors = 10
sectorType = "ONE_SECTOR"
searchNeighbourhood = arcpy.SearchNeighborhoodStandard(majSemiaxis, minSemiaxis,
angle, maxNeighbors,
minNeighbors, sectorType)

# Check out the ArcGIS Geostatistical Analyst extension license
arcpy.CheckOutExtension("GeoStats")

cellSize, searchNeighbourhood, rbf, smallscaleParam)


## Licensing Information

ArcGIS for Desktop Basic: Requires Geostatistical Analyst
ArcGIS for Desktop Standard: Requires Geostatistical Analyst
ArcGIS for Desktop Advanced: Requires Geostatistical Analyst

## How radial basis functions work

RBF methods are a series of exact interpolation techniques; that is, the surface must pass through each measured sample value. There are five different basis functions:

• Thin-plate spline
• Spline with tension
• Completely regularized spline

Each basis function has a different shape and results in a different interpolation surface. RBF methods are a special case of splines.

RBFs are conceptually similar to fitting a rubber membrane through the measured sample values while minimizing the total curvature of the surface. The basis function you select determines how the rubber membrane will fit between the values. The diagram below illustrates conceptually how an RBF surface fits through a series of elevation sample values. Notice in the cross section that the surface passes through the data values.

Being exact interpolators, the RBF methods differ from the global and local polynomial interpolators, which are both inexact interpolators that do not require the surface to pass through the measured points. When comparing an RBF to IDW (which is also an exact interpolator), IDW will never predict values above the maximum measured value or below the minimum measured value as you can see in the cross section of a transect of sample data below.

However, the RBFs can predict values above the maximum and below the minimum measured values as in the cross section below.

Optimal parameters are determined using cross validation in a similar manner as explained for IDW and local polynomial interpolation.

## When to use radial basis functions

RBFs are used to produce smooth surfaces from a large number of data points. The functions produce good results for gently varying surfaces such as elevation.

However, the techniques are inappropriate when large changes in the surface values occur within short distances and/or when you suspect the sample data is prone to measurement error or uncertainty.

## The concepts behind radial basis functions

In Geostatistical Analyst, RBFs are formed over each data location. An RBF is a function that changes with distance from a location.

For example, suppose the radial basis function is simply the distance from each location, so it forms an inverted cone over each location. If you take a cross section of the x,z plane for y = 5, you will see a slice of each radial basis function. Now, suppose you want to predict a value at y = 5 and x = 7. The value of each radial basis function at the prediction location can be taken from the figure above, given by the values Φ1, Φ2 , and Φ3, which simply depend on the distance from each data location. The predictor is formed by taking the weighted average w1Φ1 + w2Φ2 + w3Φ3 + …

Now the question is how to determine the weights? So far, you have not used the data values at all. The weights w1, w2, w3, and so on, are found by requiring that, when the prediction is moved to a location with a measured value, the data value is predicted exactly. This forms N equations with N unknowns and can be solved uniquely. Thus, the surface passes through the data values, making predictions exact.

The radial basis function in this example is a special case of the multiquadric RBF. Geostatistical Analyst also allows you to use other RBFs such as completely regularized splines, thin-plate splines, splines with tension, and inverse multiquadric. Sometimes, the difference between these is not great, but you may have reason to choose one or you can try several and use cross-validation to select one. Each of the RBFs has a parameter that controls the "smoothness" of the surface.

For all methods except inverse multiquadric, the higher the parameter value, the smoother the map; the opposite is true for inverse multiquadric.

#### {{htmlJSON.HowtoCite}}

ESRI (2019). Radial Basis Functions, Model Item, OpenGMS, https://geomodeling.njnu.edu.cn/modelItem/2c155642-6e5c-43ca-a41a-9a51c18a80aa

#### Authorship

:
View
Is authorship not correct? Feed back

#### QR Code

• {{curRelation.name}}
{{curRelation.name}}

{{curRelation.overview}}
{{curRelation.author.join('; ')}}
{{curRelation.journal}}

{{htmlJSON.RelatedItems}}
{{props.row.name}}

{{ props.row.overview }}
{{ props.row.overview }}
Drop the file here, orclick to upload.
File size should not exceed 10m.
Select From My Space

{{htmlJSON.authorshipSubmitted}}

Cancel Submit
{{htmlJSON.Cancel}} {{htmlJSON.Submit}}
{{ item.label }} {{ item.value }}
{{props.row.localName}}
{{htmlJSON.ModelName}}:
{{htmlJSON.Cancel}} {{htmlJSON.Submit}}
模型名称
名称 别名 {{tag}}

模型版本
系列名 版本号 目的 修改内容 创建/修改日期 作者

描述信息
摘要 详细描述

{{tag}}
* 时间参考系
* 空间参考系类型 * 空间参考系名称

开发信息
起始日期 终止日期 进展 开发者

* 是否开源 * 访问方式 * 使用方式 开源协议 * 传输方式 * 获取地址 * 发布日期 * 发布者

元数据版本
编号 目的 修改内容 创建/修改日期 作者
{{index+1}}

{{index+1}}

{{index+1}}

模型类型

分类信息

时间分辨率 时间尺度 时间步长 时间范围 空间维度 格网类型 空间分辨率 空间尺度 空间范围
{{tag}}
* 类型
图例

* 名称 * 描述
上传

示例描述 * 名称 * 类型 * 值/链接 上传

{{htmlJSON.Cancel}} {{htmlJSON.Submit}}
Title Author Date Journal Volume(Issue) Pages Links Doi Operation
{{htmlJSON.Cancel}} {{htmlJSON.Submit}}

Yes, this is it Cancel

OK
{{htmlJSON.Cancel}} {{htmlJSON.Confirm}}
Model Classifications 1
Model Classifications 2
Title Author Date Journal Volume(Issue) Pages Links Doi Operation

#### NEW

Name:
Affiliation:
Email:
Homepage:

Yes, this is it Cancel

Confirm
{{htmlJson.path}}
:
/{{path.name}}
search results of '{{searchContentShown}}'

#### No content to show

{{item.name}}

.

{{item.suffix}}

.{{item.suffix}}

{{htmlJson.Max}}: {{toDecimal1(capacity/1073741824)}} GB
Copy
Delete
Rename
/{{path.label}}
{{htmlJson.Change}}
/{{path.name}}
{{htmlJson.SelectFile}}
{{htmlJson.Cancel}} {{htmlJson.Confirm}}
{{htmlJson.path}}
:
/{{path.name}}
/..
{{htmlJson.Cancel}} {{htmlJson.Confirm}}
{{ data.name }}
##### You have select  {{multipleSelection.length+multipleSelectionMyData.length}} data .
• Output Data
• {{item.computableName}}@{{formatDate(item.runTime)}}
{{scope.row.type}}
{{ scope.row.tag }}
• Fork Data
{{it.category}}

#### NEW

Name:
Affiliation:
Email:
Homepage:
previous next conform
{{htmlJSON.ModelClassifications}}

{{htmlJson.RelatedItems}}
{{ props.row.overview }}
{{ props.row.overview }}
{{htmlJson.Cancel}} {{htmlJson.OK}}
{{ item.label }} {{ item.value }}
{{props.row.localName}}
Model Name :