Another Look at Non-Euclidean Variography

Transcription

1 Another Look at Non-Euclidean Variography G. Dubois European Commission DG Joint Research Centre Institute for Environment and Sustainability, Ispra, Italy. ABSTRACT: Tobler s first law of geography states: Everything is related to everything else, but near things are more related than distant things. This property, which holds for most environmental variables, is used in spatial statistics: a weighted moving average is often applied to estimate data, and the weights can be a simple function of distance or derived from a model of the spatial covariance. However, many situations in which observations are close in space but distant in a physical sense, require some means to infringe on Tobler s law. This is typically the case with measurements made in streams, as the water distance can be very different from the Euclidean distance. Exploring the use of non-euclidean spaces may thus be interesting although many fundamental theoretical and practical problems exist. If modelling of the variogram is often at stake when applied to non-euclidean spaces, we argue, however, that exploratory non-euclidean variography may provide valuable information. It is the purpose of this paper to discuss non-euclidean variography by means of a case study in which information provided by a digital elevation model (DEM) is used to explore monthly rainfall observations made at 36 meteorological stations in the Algarve region, Portugal. KEYWORDS: Exploratory variography, non-euclidean, cost-weighted distances, magnifying factor 1. Introduction Everything is related to everything else, but near things are more related than distant things. Tobler s (197) first law of geography, taken literally, would obviously generate numerous problems when dealing with observations that have very different origins but are nevertheless close in space. This is typically the case with a number of variables such as soils or geological layers that can present abrupt changes in space. Geologists in particular have quickly set up computer techniques for taking faults into account or for imposing some external influence on the contours generated by the gridding algorithms (see e.g. Zoraster, 1996). More recently, a number of authors have explored means to use water distances rather than Euclidean distances for modelling variograms and estimate data in a water network. While Rathbun (1998) and Little et al. (1997) report small changes only in their estimates when non-euclidean distances are used, Gardner et al (23), working at larger scales, observed improvements in their results. All authors, however, noticed a reduction in their nugget effect. Generally, a metric space is a set A in which a distance function or metric d is defined, with the following properties: d(a, b) = d(b, a) for all a, b A (symmetry) (1) d(a, b) for all a, b A (non-negativity) and with equality if and only if a = b; (2) d(a, b) d(a, c) + d(c, b) for all a, b, c A. (triangle inequality); (3) Liège September, 3 rd - 8 th 26 S13-8

2 Curriero (1996) and Rathbun (1998) noted that this metric space further needs to satisfy the positive definite property of the correlation function k k mm i j ρ ( d( ai, a j)) (4) i= 1 j= 1 A metric space need not be Euclidean. Curriero (1996) and Rathbun (1998) underlined that using non-euclidean distances does not guarantee that existing variogram functions remain valid. Hence, recent efforts have focused on ensuring that the non-euclidean method adopted would lead to variogram models that are theoretically valid (see Løland and Høst, 23; Cressie et al, 25; Curriero, 25). 2. Cost-weighted distances for non-euclidean transformations One may logically try to extend the concept of using non-euclidean spaces in networks to grids. Similarly to using a network topology for calculating water distances, one can derive some travel distances between cells on a grid. The computation of cost-weighted distances using a digital elevation model (DEM) can easily be calculated using a geographic information system (GIS). As an example, such an operation is frequently used to calculate travelling costs between two locations (Figure 1). Fig. 1. Computing cost-weighted distances using a Digital Elevation Model (DEM) Looking at Figure 1, one will quickly realise that, depending on how the costs are defined, the defining properties of a metric space may not hold anymore. Using the example of driving between two points a and b, the symmetry condition (Eq. 1) is not valid anymore if a is located higher than b. This can be fixed if one considers simply the geometrical properties of the DEM and the distance one would have to walk between a and b would be equal to the distance between b to a. However, the triangle inequality (Eq. 3) may be violated if there is a large barrier between two points a and b so that it would be shorter to pass by a third point c. From a geostatistical point of view, the pairwise comparisons used for modelling the variogram should not be computed using such cost-weighted distances, since the costs between cells would constantly vary between pairs and the adopted metric space would have no basis. In other words, the approach suggested above for calculating variograms will show spatial structures within spaces that do not exist, and the theoretical covariance models will not be valid! These points have been underlined by Krivoruchko and Gribov (22) who proposed the use of cost-weighted distances in combination with some alternative moving window kernel instead of a geostatistical approach. However, if we agree that modelling such non-euclidean variograms has to be excluded, one may still look back at the information provided during the exploratory variography step. The pairwise comparison files, h- Liège September, 3 rd - 8 th 26 S13-8

3 scatterplots, variogram clouds and experimental variograms (see e.g. Pannatier, 1996) are first of all means to visualise some statistics of points separated by a distance, may it be derived from a cost-weighted distance algorithm. Within this spirit, Dubois (21) explored the use of cost-weighted distances derived from a DEM to analyse the impact of the Austrian relief on the spatial distribution of 137 Cs as well as the impact of the Swiss topography on a set of daily rainfall measurements. In both case studies, some regularisation effect on the experimental variogram could be shown as well as some reduction of the nugget effect. The approach could underline the impact of the resolution of the DEM as well as of the choice of cost functions. In particular, if one uses some geometrical algorithm to calculate the walking distance between points, no differences between the experimental variograms will be seen, unless some magnifying factor is used to increase artificially the impact of the DEM. These points will be further discussed in the following case study. 3. Case study: the Algarve data set The data set analysed here consists of a DEM with resolution of 1 km 2 of the Algarve region and of rainfall measurements collected over the period January 197 to March 1995 (Figure 2). Depending on the month, the correlation between rainfall and elevation ranges from.33 to.83. The correlation between the altitude and the average annual rainfall data is.79. The readers will find detailed analyses of these data in Goovaerts (1999, 2), who discussed various ways to incorporate information from a DEM when predicting rainfall fields. Fig. 2. Proportional symbol maps of the average annual rainfall measurements (red dots) on top of a DEM (1 km 2 resolution) of Algarve, Portugal. As already mentioned, using some walking path distance between pairs of points to take the elevation into account will not show significant changes in the experimental variograms unless the signal of the auxiliary variable (here the DEM) is amplified. This can easily be done by multiplying the elevations by any value. For illustration purposes, the DEM has been multiplied by values of 1, 1, 1 and 1. The choice of such magnifying factors is empirical and will depend on the nature of the variables and the spatial resolution of the data. Here, magnifying factors below 1 will not reveal a clear impact of the use of the weighted distances, and a value of 1 will not show differences from the picture obtained with a factor 1. Figure 3 shows box plots of the squares of the differences calculated for all the 63 pairs of points (z(x)-z(x i +h)) of the annual rainfall data. The distances between z(x) and z(x i +h) shown have been calculated using the Euclidean distances as well as weighted distances using magnifying factors of 1, 1 and 1. Histograms that indicate the number of points found in each lag are also provided. Liège September, 3 rd - 8 th 26 S13-8

4 2 Untransformed space Society for Mathematical Geology Mean Mean±SE Mean±SD 18 Square of differences (z(x i)-z(xi+h) Separation distance (km) Magnifying factor 1 Mean Mean±SE Mean±SD Separation distance, km 16 9 Square of differences (z(x i)-z(xi) 2 ) Z tranformed separation distance (km) Magnifying factor 1 Mean Mean±SE Mean±SD Separation distance, Z transformed (x1), km Square of differences (z(x i)-z(xi+h) Z transformed separation distance (km) Magnifying factor 1 Mean Mean±SE Mean±SD Separation distance, Z transformed (x1), km 1 Square of differences (z(x i)-z(xi+h)) Z transformed separation distance (km) Separation distance, Z transformed (x1), km Fig. 3. Box plots (left) of the squares of the differences (z(x)-z(x i +h)) of annual rainfall data using weighted distances derived from a DEM. The DEM effect has been amplified using magnifying factors of 1, 1, 1 and 1 (top to bottom). The histograms (right) show the number of pairs of points used to calculate each box. Liège September, 3 rd - 8 th 26 S13-8

5 Figure 3 shows clearly the impact of using weighted distances on the points that are located close in space: moving farther apart, points separated by a barrier defined by the DEM will show lower mean values of the squares of differences as well as a lower standard deviation. The regularization effect of the spatial transformation is particularly visible when using a magnifying factor of 1. The approach shown above could thus be used to verify that the hypothesis of a DEM impact on the annual rainfall values is true. One still needs to be aware that the transformed distances will not allow a direct comparison of the box plots (and consequently of experimental variograms) and will so render difficult the physical interpretation of the observations. On the other hand, the number of pairs of points used to calculate the mean values and associated standard deviations found for each lag clearly allows us to show how the variability between measurements found within similar ranges has been reduced. Figure 4 shows some further tentative to illustrate the impact of cost-weighted distances. Euclidean and non-euclidean experimental variograms have been calculated using average values for the rainy month of December (left) and the dry month of July (right). The correlation between rainfall values and altitude are.71 and.39, respectively. The magnifying factor used here is only 2. This value was found to be a good compromise between the need to reveal the impact of the spatial transformation on the variograms and the necessity to render the lag distances almost comparable. Semivariance Euclidean Non Euclidean Semivariance Lag (x 5 km).2 Euclidean Non Euclidean Lag (x 5 km) Fig. 4. Non-Euclidean experimental variograms for the months of December (left) and July (right). Magnifying factor: DEM x 2. If the DEM does not show much effect at short scale for the month of July (the variogram shows anyway that the measurements are not correlated in space), it does show a clear influence on the semivariance for the month of December. Again, taking into account the barrier effect of the relief makes it easier to regularize the spatial correlation of the rainfall data. 4. Conclusions The purpose of this case study was to discuss the potential use of transforming spaces for exploratory data analysis. Researchers may, using cost-weighted distances, find simple means to verify the existence of a potential barrier effect on their data as well as new means to identify outliers better (using non-euclidean h-scatterplots or variogram clouds for example). The paper further underlines the frequent need to magnify the impact of the auxiliary variable Liège September, 3 rd - 8 th 26 S13-8

6 on the primary variable to see any effect. Many issues remain unexplored such as the empirical definition of the weights used for distance transformation as well as the physical interpretation of these non-euclidean experimental variograms. Regarding the use of such transformation for estimation purposes through geostatistics, the issue remains unresolved because of the lack of valid covariance models. Still, using a simple inverse distance approach for cross-validation purposes will show a (small but systematic) reduction in the average relative mean errors when non-euclidean spaces are used with the Algarve data sets. This cross-validation approach may be computationally intensive as it requires from the algorithm to recalculate for each point the new weighted distances to all the neighbouring points. Acknowledgments: The z-distance algorithm developed for Surfer was written by Aleksey Amantov. The script is available from in the download section. The Algarve data set was provided by Pierre Goovaerts. Tore Tollefsen has proofread this paper. The author acknowledges their kind support. REFERENCES Curriero, F. C. (1996). The Use of Non-Euclidean Distances in Geostatistics. PhD Dissertation, Department of Statistics, Kansas State University. Curriero, F. C. (25). On the use of non-euclidean isotropy in geostatistics. The Berkeley Electronic Press, Collection of Biostatistics Research Archive, Paper 94, 29 pp. Dubois, G. (21). Intégration de Systèmes d Informations Géographiques (SIG) et de méthodes géostatistiques. Ph. D. Dissertation, Department of Earth Sciences, University of Lausanne (in French). Gardner, B., Sullivan, P.J. and A. J. Lembo. (23). Predicting stream temperatures: geostatistical. model comparison using alternative distance metrics. Canadian Journal of Fisheries and Aquatic Sciences, (6): Goovaerts, P. (1999). Performance comparison of geostatistical algorithms for incorporating elevation into the mapping of precipitation. In: Proceedings of Geocomputation 99. Goovaerts, P. (2). Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall. Journal of Hydrology (228): Krivoruchko, K. and A. Gribov (22). Geostatistical interpolation in the presence of barriers. Available from ESRI online: Little, L., Edwards, D. and D. E. Porter (1997). Kriging in estuaries: as the crow flies, or as the fish swims?, Journal of Experimental Marine Biology and Ecology, (213): Løland, A., and Høst, A. (23). Spatial covariance modelling in a complex coastal domain by multidimensional scaling. Environmetrics (14): Rathbun, S. L. (1998). Spatial modelling in irregularly shaped regions: kriging estuaries. Environmetrics (9): Pannatier, Y. (1996). VARIOWIN: Software for Spatial Data Analysis in 2D. Springer-Verlag, New York. Tobler, W. (197). A computer movie simulating urban growth in the Detroit region. Economic Geography. (46): Zoraster, S. (1996). Imposing geologic interpretations on computer-generated contours using distance transformations. Mathematical Geology (28): Cressie, N., Frey, J., Harch, B. and M. Smith (25). Spatial Prediction on a River Network. Technical Report No. 747, Department of Statistics, The Ohio State University. Liège September, 3 rd - 8 th 26 S13-8