Interpolation par recherche d’information locale

Interpolation by local information

¹ Laboratoire ThéMA, Université de Franche-Comté, Besançon
² Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon
³ CESAER, INRA, Dijon

^* daniel.joly@univ-fcomte.fr

Résumé

Les méthodes d’interpolation, qu’elles procèdent de régressions ou de krigeage, sont globales, en ce sens qu’elles utilisent l’ensemble des données disponibles sur un territoire donné. Or, quand l’hétérogénéité spatiale de celui-ci est forte, la qualité des résultats s’en ressent. Pour dépasser cette difficulté, une méthode d’interpolation locale est proposée et appliquée, à titre de test, aux températures du territoire français. Partant d’un jeu de stations de mesures réparties sur cet espace digitalisé à 250 m, la méthode consiste à modéliser les variations spatiales locales de la température en considérant chaque point de la grille et les n stations voisines les plus proches qui l’entourent. La procédure enchaîne les étapes suivantes : reconnaissance des n stations les plus proches du point d’estimation et partition du territoire en polygones définis selon une règle de voisinage, établissement d’un modèle local par régression multiple pour chaque polygone, application des coefficients obtenus de la régression en vue d’obtenir une valeur estimée en chaque point du polygone considéré. Ces résultats sont comparés à ceux de trois méthodes d’interpolation globales : (i) régressions, (ii) krigeage ordinaire, (iii) régressions avec krigeage des résidus. Une quatrième partie développe les résultats originaux autorisés par l’interpolation locale : cartographie du coefficient de détermination, du coefficient de la corrélation température/altitude et de la constante (intercept) liée à l’altitude. Avec ces développements, l’accent est d’abord mis sur les processus qui commandent la variation spatiale du climat.

Abstract

The paper describes a local interpolation method applied to temperature and sunshine in France. First, the interpolation is based on many geographical variables which are derived from a Digital Elevation Model with a 250 m resolution (gradient, aspect, topographical rugosity, theoretical quasi global radiation energy computed for a cloudless sky on an equinox day, aso), and from a land cover image (distance to sea or ocean, distance to forest, etc.). All these data layers are stored in a GIS as raster files. Second, three main methods are used to interpolate climatic data and obtain a continuous spatial field: (i) deterministic method based on regressions, (ii) geostatistic method (kriging) and (iii) regressions and kriging the residuals in the same process. All three are solved using all the available data as a whole: 651 stations for monthly mean temperature and 111 stations recording monthly duration of sunshine. The quality of the estimations varies from one method to an other: for example, in July, the standard deviation for the three interpolation methods is (i) 1.61°C, (ii) 0.92°C and (iii) 0.72°C. All of them generate 2 or 3% of huge residuals (down to -4°C and up to 4°C). The statistics generally fail for stations located in particular places (for example, station having a steep slope, among many others located on flat areas). In such a case, the local conditions which occur there are not taken into account by the global model.

That is the reason why a fourth method based on local interpolations was tested. The principle of the method is similar to the one named ‘local regressions’ by Loader (2004). It consists of researching, around a cell to be estimated, the n closest climatic stations. The value given to the parameter n is significant. The higher n is, the wider the area covered by the stations is; on the other hand, the higher n is, the better the statistic are but the longer the time computing is. Considering these constraints, the value n=30 was chosen. Thus, the n stations makes up the data set for which temperature is estimated by multiple regressions (temperature is the explained variable, the layers stored in the GIS are used as explanatory variables). Finally, the regression model is taken as a cartographic operator. This modelling process is repeated, pixel by pixel, until temperature distribution is calculated for the whole study area. To speed the calculations up, the study area is segmented into many thousands of polygons according to a proximity rule: all the cells belonging to a polygon refer to the same n stations. Thus, a single multiple regression is processed as part of each polygon and the obtained multiple regression coefficients are applied to all the cells of it. Let us notice that, from a geographical point of view, the number and the size of polygons vary according to n.

The local interpolation method improves the quality of estimation: it reduces both the standard deviation (0.38°C for July) and the occurrences of high residuals. Contrary to the other methods, the local interpolation method allows to produce additional maps which are highly informative. Then, other and highly informative maps can be delivered. As examples, three maps for July are given: determination coefficients, correlation temperature / elevation coefficients, temperature/elevation intercepts.