1,720,993 research outputs found
Mapping precipitation in Switzerland with ordinary and indicator kriging. Special issue: Spatial Interpolation Comparison 97
No full textDesigning optimal sampling configurations with ordinary and indicator kriging
No full textThe objective of this paper is to examine the applicability of two geostatistical approaches, ordinary kriging (OK) and ordinary indicator kriging (IK), to the design of optimal sampling strategies. A by-product of OK is the OK variance. The OK variance is a measure of confidence in estimates. It is a function of (i) the form of spatial variability of the data (modelled, for example, by the variogram), and (ii) the spatial configuration of the samples. The disadvantage of the OK variance is that it is independent of the magnitude of data values locally. For a given sampling configuration, the OK variance will be the same irrespective of the data values; thus, if data are measured on a regular grid, the maximum OK variance will be identical; however, much of the data vary locally. An approach that is conditional on the data values would be more suitable in such cases. This paper uses the conditional variance of the conditional cumulative distribution function (ccdf) derived through IK to assess local uncertainty in estimates. Since the conditional variance of the ccdf is conditional on the data values, the problem of OK variance data independence is overcome. Previously, to determine an acceptable sample grid spacing, investigators have plotted the maximum OK variance for a range of sample spacings and used the plot to select a sample spacing that achieved a given precision of estimation; however, where the spatial variability is not stationary across the region of concern, the OK variance will be biased as it is independent of the data values locally. The maximum conditional variance was used in the same way to account for the magnitude of data values as well as the form of spatial variability and the spatial configuration of the data. A photogrammetically derived digital terrain model (DTM) was sampled on a regular grid, and the success of the OK and IK approaches in ascertaining optimal sampling intervals was examined and compared with reference to the DTM. Once the variogram and indicator variograms were computed for the sample data, mathematical models were fitted and the model coefficients were used for kriging. The performance of the two approaches was assessed in three separate ways: (i) The model coefficients were used to ascertain the maximum OK and conditional variance for several sampling intervals; (ii) The DTM was then sampled at several (progressively smaller) spacings, and estimates were made from the samples. The differences between the estimates and the population (that is, the complete DTM) were then computed and the errors using OK and IK were related to the maximum error that was predicted by the OK variance or the conditional variance. The proportion of estimates that fell outside the estimation variances were quantified and the different results compared; (iii) The estimation errors for each grid cell were plotted against the OK variance and conditional variance, and the form of the relationships was assessed. Finally, the implications of using the two approaches were discussed
Scale and the spatial structure of landform: optimising sampling strategies with geostatistics
No full textInformation on the scale or frequency of spatial variation in properties such as land-form is of value in a wide variety of contexts including classification of land-form types and as an input to environmental modelling applications. This paper utilises this information to demonstrate how producers of digital terrain data sets may ascertain the best approach to employ and the nature and configuration of data that would be required to fulfil a particular user's requirements in terms of information and accuracy. The approach presented is applicable whether data are sampled on the ground or by remote sensing. The research centres around an examination of Ordnance Survey(R) Land-Form PROFILE (TM) contour data and the particular focus is the potential of the application of geostatistics to the improvement of the National Height Dataset (NHD). The research illustrates that to apply geostatistics it was necessary to ensure homogeneity of spatial variation across the region in concern. That is, it was necessary to classify spatial variation. Once classification of spatial variation was achieved it was possible to quantify the variation (using the variogram) and identify dominant forms of spatial variation for particular regions to aid the design of optimal sampling strategies. The results demonstrate that significant gains in efficiency can be obtained by adapting (i) the geostatistical approach and (ii) classifying the spatial variation prior to the application of geostatistics. <br/
Deriving ground surface digital elevation models from LiDAR data with geostatistics
No full textThis paper focuses on two common problems encountered when using Light Detection And Ranging (LiDAR) data to derive digital elevation models (DEMs). Firstly, LiDAR measurements are obtained in an irregular configuration and on a point, rather than a pixel, basis. There is usually a need to interpolate from these point data to a regular grid so it is necessary to identify the approaches that make best use of the sample data to derive the most accurate DEM possible. Secondly, raw LiDAR data contain information on above-surface features such as vegetation and buildings. It is often the desire to (digitally) remove these features and predict the surface elevations beneath them, thereby obtaining a DEM that does not contain any above-surface features. This paper explores the use of geostatistical approaches for prediction in this situation. The approaches used are inverse distance weighting (IDW), ordinary kriging (OK) and kriging with a trend model (KT). It is concluded that, for the case studies presented, OK offers greater accuracy of prediction than IDW while KT demonstrates benefits over OK. The absolute differences are not large, but to make the most of the high quality LiDAR data KT seems the most appropriate technique in this case
Characterising spatial variability and assessing uncertainty in a photogrammetrically derived digital terrain model with variogram and the fractal dimension
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Increased accuracy of geostatistical prediction of nitrogen dioxide in the United Kingdom with secondary data
No full textFive techniques were used to map nitrogen dioxide (NO2) concentrations in the United Kingdom. The methods used to predict from point data, collected as part of the UK NO2 diffusion tube network, were local linear regression (LR), inverse distance weighting (IDW), ordinary kriging (OK), simple kriging with a locally varying mean (SKlm) and kriging with an external drift (KED). These techniques may be divided into two groups: (i) those that use only a single variable in the prediction process (IDW, OK) and (ii) those that make use of additional variables as a part of prediction (LR, SKlm and KED). Nitrous oxides emission data were used as secondary data with LR, SKlm and KED. It was concluded that SKlm provided the most accurate predictions based on the summary statistics of prediction errors from cross-validation. <br/
Non-stationary approaches for mapping terrain and assessing uncertainty
No full textIt is well known that terrain may vary markedly over small areas and that statistics used to characterise spatial variation in terrain may be valid only over small areas. In geostatistical terminology, a non-stationary approach may be considered more appropriate than a stationary approach. In many applications, local variation is not accounted for sufficiently. This paper assesses potential benefits in using non-stationary geostatistical approaches for interpolation and for the assessment of uncertainty in predictions with implications for sampling design. Two main non-stationary approaches are employed in this paper dealing with (1) change in the mean and (2) change in the variogram across the region of interest. The relevant approaches are (1) kriging with a trend model (KT) using the variogram of residuals from local drift and (2) locally-adaptive variogram KT, both applied to a sampled photogrammetrically derived digital terrain model (DTM). The fractal dimension estimated locally from the double-log variogram is also mapped to illustrate how spatial variation changes across the data set. It is demonstrated that estimation of the variogram of residuals from local drift is worthwhile in this case for the characterisation of spatial variation. In addition, KT is shown to be useful for the assessment of uncertainty in predictions. This is shown to be true even when the sample grid is dense as is usually the case for remotely-sensed data. In addition, both ordinary kriging (OK) and KT are shown to provide more accurate predictions than inverse distance weighted (IDW) interpolation, used for comparative purposes
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