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Analyse Spatiale (Interpolation)

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Objectifs :

Understanding of interpolation as part of spatial analysis

Mots clés :

Point data, interpolation method, Inverse Distance Weighted, Triangulated Irregular Network

Aperçu

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.

L’Interpolation spatiale en détail

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:

../../_images/temperature_map.png

Carte de température interpolée avec les stations météo d’Afrique du Sud.

Because of high cost and limited resources, data collection is usually conducted only in a limited number of selected point locations. In GIS, spatial interpolation of these points can be applied to create a raster surface with estimates made for all raster cells.

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.

Il existe plusieurs méthodes d’interpolation. Dans cette introduction, nous présenterons deux méthodes largement utilisées, à savoir Pondération par Distance inverse (IDW) et **Interpolation Triangulaire**(TIN). Si vous recherchez des méthodes d’interpolation supplémentaires, veuillez vous reporter à la section ‘Pour aller plus loin’ à la fin de cette rubrique.

Pondération par l’Inverse de la Distance (IDW)

In the IDW interpolation method, the sample points are weighted during interpolation such that the influence of one point relative to another declines with distance from the unknown point you want to create (see figure_idw_interpolation).

Figure IDW Interpolation 1:

../../_images/idw_interpolation.png

Inverse Distance Weighted interpolation based on weighted sample point distance (left). Interpolated IDW surface from elevation vector points (right). Image Source: 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.

In GIS, interpolation results are usually shown as a 2 dimensional raster layer. In figure_idw_result, you can see a typical IDW interpolation result, based on elevation sample points collected in the field with a GPS device.

Figure IDW Interpolation 2:

../../_images/idw_result.png

IDW interpolation result from irregularly collected elevation sample points (shown as black crosses).

Triangulated Irregular Network (TIN)

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:

../../_images/tin_interpolation.png

Delaunay triangulation with circumcircles around the red sample data. The resulting interpolated TIN surface created from elevation vector points is shown on the right. Image Source: 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:

../../_images/tin_result.png

Delaunay TIN interpolation result from irregularly collected rainfall sample points (blue circles)

Problèmes courants / Choses à savoir

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. Evaluate the sample data. Do this to get an idea on how data are distributed in the area, as this may provide hints on which interpolation method to use.
  2. Apply an interpolation method which is most suitable to both the sample data and the study objectives. When you are in doubt, try several methods, if available.
  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.

Autres méthodes d’interpolation

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.

Qu’avons-nous appris?

Faisons le point sur ce que nous avons abordé dans cette partie:

  • Interpolation uses vector points with known values to estimate values at unknown locations to create a raster surface covering an entire area.
  • The interpolation result is typically a raster layer.
  • It is important to find a suitable interpolation method to optimally estimate values for unknown locations.
  • IDW interpolation gives weights to sample points, such that the influence of one point on another declines with distance from the new point being estimated.
  • TIN interpolation uses sample points to create a surface formed by triangles based on nearest neighbour point information.

Maintenant, essayez !

Voici quelques pistes d’actions à essayer avec vos élèves:

  • Le ministère de l’Agriculture souhaite cultiver de nouvelles terres dans votre région mais, mis à part les caractéristiques des sols, il souhaite savoir si la pluviométrie sera suffisante pour assurer un bon rendement. Toutes les informations dont il dispose proviennent des stations météo de la région. Créer une surface interpolée avec vos étudiants qui délimite les zones recevant le plus de précipitations.

  • 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?

Something to think about

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.

Pour aller plus loin

Livres :

  • 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.

Sites Web :

Le Guide de l’Utilisateur de QGIS contient également plus d’informations détaillées sur les outils d’interpolation avec QGIS.

La suite ?

This is the final worksheet in this series. We encourage you to explore QGIS and use the accompanying QGIS manual to discover all the other things you can do with GIS software!