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Spatial analysis is the process of manipulating spatial information to extract new information and meaning from the original data. Usually spatial analysis is carried out with a Geographic Information System (GIS). A GIS usually provides spatial analysis tools for calculating feature statistics and carrying out geoprocessing activities as data interpolation. In hydrology, users will likely emphasize the importance of terrain analysis and hydrological modelling (modelling the movement of water over and in the earth). In wildlife management, users are interested in analytical functions dealing with wildlife point locations and their relationship to the environment. Each user will have different things they are interested in depending on the kind of work they do.


Spatial interpolation is the process of using points with known values to estimate values at other unknown points. For example, to make a precipitation (rainfall) map for your country, you will not find enough evenly spread weather stations to cover the entire region. Spatial interpolation can estimate the temperatures at locations without recorded data by using known temperature readings at nearby weather stations (see figure_temperature_map). This type of interpolated surface is often called a statistical surface. Elevation data, precipitation, snow accumulation, water table and population density are other types of data that can be computed using interpolation.

Figure Tempareature Map 1:



高コストと限られた資源によって、データ収集は通常、選択されたポイントの位置の限られた数で行われる。 GISにおいては、これらの点の空間補間は、全てのラスタのセルに対して行われた見積りでラスタサーフェスを作成するために適用することができる。

In order to generate a continuous map, for example, a digital elevation map from elevation points measured with a GPS device, a suitable interpolation method has to be used to optimally estimate the values at those locations where no samples or measurements were taken. The results of the interpolation analysis can then be used for analyses that cover the whole area and for modelling.

There are many interpolation methods. In this introduction we will present two widely used interpolation methods called Inverse Distance Weighting (IDW) and Triangulated Irregular Networks (TIN). If you are looking for additional interpolation methods, please refer to the ‘Further Reading’ section at the end of this topic.


IDW補間法では、作成したい不明な点からの距離に影響を受けて、サンプル点の補間中に別の離れた1点からの距離で重み付けされるようにします  (figure_idw_interpolation を参照)。

Figure IDW Interpolation 1:


逆距離加重補間は重み付けられたサンプル点の距離に基づきます(左)。標高ベクタポイントから補間IDWサーフェス(右)。イメージソース: Mitas, L., Mitasova, H. (1999).

Weighting is assigned to sample points through the use of a weighting coefficient that controls how the weighting influence will drop off as the distance from new point increases. The greater the weighting coefficient, the less the effect points will have if they are far from the unknown point during the interpolation process. As the coefficient increases, the value of the unknown point approaches the value of the nearest observational point.

It is important to notice that the IDW interpolation method also has some disadvantages: the quality of the interpolation result can decrease, if the distribution of sample data points is uneven. Furthermore, maximum and minimum values in the interpolated surface can only occur at sample data points. This often results in small peaks and pits around the sample data points as shown in figure_idw_interpolation.

GISにおいて、補間結果は、通常、2次元ラスタ層として示されています。 figure_idw_result では、GPSデバイスとでフィールド収集された標高サンプル点に基づいて、典型的なIDW補間結果を見ることができます。

Figure IDW Interpolation 2:




TIN interpolation is another popular tool in GIS. A common TIN algorithm is called Delaunay triangulation. It tries to create a surface formed by triangles of nearest neighbour points. To do this, circumcircles around selected sample points are created and their intersections are connected to a network of non overlapping and as compact as possible triangles (see figure_tin_interpolation).

Figure TIN Interpolation 1:


赤のサンプルデータの周りの外接円とのドロネー三角形分割。標高ベクタポイントから作成された結果として補間したTINサーフェスを右側に表示します。イメージソース: Mitas, L., Mitasova, H. (1999).

The main disadvantage of the TIN interpolation is that the surfaces are not smooth and may give a jagged appearance. This is caused by discontinuous slopes at the triangle edges and sample data points. In addition, triangulation is generally not suitable for extrapolation beyond the area with collected sample data points (see ).

Figure TIN Interpolation 2:



一般的な問題 / 注意すべき点

It is important to remember that there is no single interpolation method that can be applied to all situations. Some are more exact and useful than others but take longer to calculate. They all have advantages and disadvantages. In practice, selection of a particular interpolation method should depend upon the sample data, the type of surfaces to be generated and tolerance of estimation errors. Generally, a three step procedure is recommended:

  1. サンプルデータを評価します。これは補間方法を使用するかについてのヒントを提供することができるように、データが地域に分布している方法についてのアイデアを得るためにこれを行います。

  2. サンプルデータと研究目的の両方に最適な補間法を適用します。疑問がある場合には利用可能であれば、いくつかの方法を試してみてください。

  3. Compare the results and find the best result and the most suitable method. This may look like a time consuming process at the beginning. However, as you gain experience and knowledge of different interpolation methods, the time required for generating the most suitable surface will be greatly reduced.


Although we concentrated on IDW and TIN interpolation methods in this worksheet, there are more spatial interpolation methods provided in GIS, such as Regularized Splines with Tension (RST), Kriging or Trend Surface interpolation. See the additional reading section below for a web link.



  • **補間**は 全領域をカバーするラスタサーフェスを作成することを目的とし未知の場所で値を推定するために、既知の値を持つベクタポイントを使用しています。

  • 補間結果は典型的な **ラスタ**レイヤです。

  • 最適未知の場所の値を推定するための**適切な補間法を見つける**ことが重要です。

  • IDW補間法では、新しく見積もられた別の離れた新点からの距離の影響を受けたサンプル点に重みを与える手法です。

  • TIN 補間法 は最寄りのポイント情報に基づいた三角形で形成されたサーフェスを作成するためのサンプル点を使います。



  • The Department of Agriculture plans to cultivate new land in your area but apart from the character of the soils, they want to know if the rainfall is sufficient for a good harvest. All the information they have available comes from a few weather stations around the area. Create an interpolated surface with your learners that shows which areas are likely to receive the highest rainfall.
  • The tourist office wants to publish information about the weather conditions in January and February. They have temperature, rainfall and wind strength data and ask you to interpolate their data to estimate places where tourists will probably have optimal weather conditions with mild temperatures, no rainfall and little wind strength. Can you identify the areas in your region that meet these criteria?


If you don’t have a computer available, you can use a toposheet and a ruler to estimate elevation values between contour lines or rainfall values between fictional weather stations. For example, if rainfall at weather station A is 50 mm per month and at weather station B it is 90 mm, you can estimate, that the rainfall at half the distance between weather station A and B is 70 mm.



  • Chang, Kang-Tsung (2006). Introduction to Geographic Information Systems. 3rd Edition. McGraw Hill. ISBN: 0070658986
  • DeMers, Michael N. (2005): Fundamentals of Geographic Information Systems. 3rd Edition. Wiley. ISBN: 9814126195
  • Mitas, L., Mitasova, H. (1999). Spatial Interpolation. In: P.Longley, M.F. Goodchild, D.J. Maguire, D.W.Rhind (Eds.), Geographical Information Systems: Principles, Techniques, Management and Applications, Wiley.


QGISユーザーガイドでは, QGISで提供される補間ツールの詳細な情報が含まれています.