International Journal of Remote Sensing Vol. 27, No. 16, 20 August 2006, 3375–3386
Improving satellite images classification using remote and ground data integration by means of stochastic simulation JULIA CARVALHO*, AMILCAR SOARES and ANA BIO Environmental Group of the Centre for Modelling Petroleum Reservoirs, CMRP, Instituto Superior Te´cnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal (Received 24 November 2004; in final form 24 February 2006 ) A methodology is proposed, to assess land surface cover classification using a geostatistical methodology of stochastic simulation, direct sequential cosimulation, to combine field observations with remotely sensed data classified with the classical algorithm of maximum likelihood classification. This procedure has two main advantages: (1) incorporation of a spatial continuity statistics; and (2) integration of different scales of information, contained in polygons (training areas) and point information (field observations), which also involves different qualities of information that is less reliable and more reliable, respectively. Moreover, this methodology allows production not only of a classified map, but also of maps of occupation proportions and of uncertainty for each thematic class. Local co-regionalization models are applied to account for local differences in both field data availability and distribution, and the correlation between these hard data and the classified satellite images as soft data. The methodology is based on two criteria: the influence of the hard data dependent on their availability and proportional to their proximity; and the influence of the soft data dependent on their local correlation to the hard data. The method is applied to a study of four economically important forest tree species on the Setu´bal Peninsula (south of Lisbon, Portugal). The results show more contiguous forest covers, i.e. more spatial contiguity, than the classical classification. In comparison to a contemporary field inventory, the proposed method improved forest cover estimations, showing a difference of only 3%.
1.
Introduction
To produce land surface cover maps, field observations and measurements are very reliable data but costly and therefore often scarce, most of the time too scarce to adequately estimate resources for large areas. Remotely sensed data, on the other hand, are a low-cost, abundant and widespread source of information, but with uncertainty associated with the sensoring and inferring processes from satellite sensor data. Compared with the more reliable field data, hard data, the more uncertain though more abundant classified remotely sensed data can be denoted as soft data. Comparison of classified (using classical methods) remotely sensed data with forest coverage records collected in the field, i.e. reliable hard data, will reveal classification errors. Classical classification is applied on a pixel-by-pixel basis, yielding an often too scattered image of land cover, as spatial continuity between neighbouring pixels is *Corresponding author. Email:
[email protected] International Journal of Remote Sensing ISSN 0143-1161 print/ISSN 1366-5901 online # 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/01431160600658099
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disregarded. Proximate pixels are likely to be similar, and this can be formalized in a modelled variogram and utilized to increase classification accuracy. The combination of scarce, but reliable, hard data with abundant, though not so reliable, soft data in a geostatistical framework is, therefore, an adequate solution to incorporate the advantages of the two types of information, in order to increase the accuracy of land surface cover maps. Taking into account the way the field data and classified satellite sensor data are spatially co-related (co-regionalization model), the idea of integrating both variables to generate another, better classification is basically simple (Soares et al. 1997, Pereira 1998): to re-classify one pixel close to a hard (field) observation, the influence of this observation prevails over the remote sensored data; the influence of the remote sensored data will prevail in the classification of any pixel far from the field data. A regional co-regionalization model based on the correlation between the two variables for the entire region (Goovaerts 2000) has its drawbacks, as the same spatial model, i.e. identical cross-variograms and cross-variances, may not be valid for the whole region. To overcome this problem, Pereira et al. (2000) proposed co-located co-kriging with local co-regionalization models. Geostatistical techniques that use spatial information for classification can be split into two distinct main groups. In the first, spatial information is used to provide data on texture; in the second, it is used to smooth the classified image (Atkinson and Lewis 2000). However, these methods are based on kriging and do not include ground data information. Given the nature of our data—forest species covers on distinct, bounded areas— kriging estimates would result in unrealistic, smooth surfaces. Simulation is likely to perform more accurately and is, furthermore, able to produce uncertainty measures of classification. Recently, Delgado et al. (2004) proposed a Landsat-SPOT digital images integration using geostatistical cosimulation techniques. The proposed methodology constitutes a classified satellite images classification that uses ground data to improve the process. The classification is based on the stochastic direct sequential cosimulation (CODSS) procedure proposed by Soares (2001), and on local co-regionalization models. It is based on two simultaneously applied criteria: the influence of the field observations is dependent on field-data availability and proportional to field data proximity (i.e. the influence of field observations on species cover estimates is greatest at field sample locations, decreasing with increasing distance to these locations); the influence of the soft data is dependent on their local correlation to the hard data (i.e. it is higher in regions of high correlation than in regions of low correlation). The procedure is based on a classical supervised maximum likelihood (ML) classification (Lillesand and Kiefer 1994, Schowengerdt 1997) of Landsat 7 ETM + data into forest species and other covers. This constitutes a first forest cover image that will function as secondary variable for the geostatistical calibration process based on forest–cover field observations. Geostatistical simulation will reproduce a spatial pattern similar to that of the classified satellite sensor data, but following the statistics of the ‘‘reliable’’ field data, thus resulting in an improvement of the original ML classification. 2.
Experimental data
The study area is the Setu´bal Peninsula (south of Lisbon, Portugal), covering about 154,000 ha of forested and bare mountainous grounds and urban areas. The hard data consist of 70 field observations (figure 1) of forest species cover proportions
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with records of eucalyptus (Eucalyptus globulus), umbrella pine (Pinus pinea), maritime pine (Pinus pinaster) and cork oak (Quercus suber), among others, collected from 2000 to 2002. The soft data are posterior probabilities for the same cover classes and for the sum of all other cover classes obtained from an assisted ML classification of a Landsat 7 ETM + scene (on a 30630 m grid) from June 2000, using a set of 214 training areas (of variable size, covering 21 different thematic classes) collected in the same period, and post–classification smoothing by a 363 grid-cell majority filter (figure 1). The classified map and training areas extend beyond the study area; the 156 training areas within the study area (table 1) were selected to validate the forest cover classification calibration obtained through CODSS. The posterior probabilities obtained from the ML classification algorithm for each of the original 21 cover classes were scaled to unit sum, producing a forest cover proportion map comparable to the field observation data, also expressed in terms of land surface cover proportions. All the thematic classes, except for the four forest species classes, were aggregated into one class, hereafter denominated ‘‘other’’. The forest cover classification is summarized in table 1. Posterior probabilities were upscaled and averaged to the 90690 m grid used throughout this study, and each new grid cell was assigned to the most frequent class of the nine underlying 30 m630 m grid cells. Apart from unavoidable errors in the presence of equally frequent competing classes, this led to a penalization of the less
Figure 1. Proportion of land covered by cork oak obtained through maximum likelihood classification of satellite sensor data (left) and observed in the field (right). Table 1. Land cover of the study area according to the classified satellite sensor data with the maximum likelihood classifier and area of the used training data. Class Eucalypt Umbrella pine Maritime pine Cork oak Other Sum
Coverage (ha)
Training areas (ha)
10460 4654 10410 26741 101570 153835
87 56 48 64 162 418
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frequent and more scattered cover classes. Therefore, eucalyptus, umbrella pine, maritime pine and cork oak had their cover percentages reduced to 6.2, 2.4, 5.7 and 16.1, respectively, while the ‘‘other’’ cover percentage increased to 69.7. Comparing the results of the ML classification to an inventory contemporary to the hard and soft datasets (Project LIFE ENV/P/00556 2002), based on field work and airphoto interpretation, the global values of land surface coverage obtained for each class considered in this study are very similar. Hence, the ML classification can be considered a good global estimator of forest coverage. This classification, upscaled to a 90 m690 m grid resolution, will be used for comparison with the classification results obtained using CODSS (figure 5). 3.
Geostatistical stochastic simulation
Direct sequential simulation (DSS) and cosimulation (CODSS) are geostatistical stochastic simulation algorithms, proposed by Soares (2001). DSS allows for the simulation of untransformed continuous variables, using local simple kriging estimates of the variable’s mean and variance to sample from the global cumulative distribution function. Analogously, CODSS allows joint simulation of several variables without previous transformation, which is an advantage over the sequential gaussian or indicator simulation algorithms. The application of geostatistical simulation techniques, in particular, CODSS, allows us to obtain simulated maps of the forest coverage, derived from the field observations (hard data) and their correlation with the soft data. This technique allows the generation of several realizations, all equally probable, preserving the basic statistical characteristics of the field data and using information derived from posterior probabilities, obtained from the ML classification, based on a co-regionalization model that reproduces the local correlation between these two variables. The calibration of ML classification data with the field observations is thus equivalent to conditioning the soft map to the spatial localization and value of the hard data. Considering Z(x) an image attribute Z at a spatial location x, a stochastic simulation algorithm reproduces, in the simulated image, the variability of a continuous variable by means of two statistics: the distribution function of Z(x)2FZ(x)5p{Z(x),x}, which ensures the frequency of the histogram classes, and the variogram c(h), which reproduces the spatial continuity of Z(x). In short, the simulated ZS(x) must satisfy the following: 1. For any value z: p{Z(xa),z}5p{ZS(x),z}; 2. c(h)5cS(h), where c(h) and cS(h) are the variograms of the field observations and simulated values, respectively; 3. ZS(xa)5Z(xa), i.e. for a field observation xa the value of Z(xa) and that of the simulated value ZS(xa) are the same. Suppose two variables, Z1(x) and Z2(x), where Z1(x) is the primary variable (in our case study, the hard data), and Z2(x) the secondary variable (i.e. the soft data), the joint simulation algorithm can be described in the following steps: 1. Define a random path visiting each node of a regular grid of nodes. 2. At each node xu simulate the value z1s(xu) using the DSS algorithm:
N
identify the local mean and variance of z1(x) as the simple kriging (SK) estimate z1(x)* and estimation variance s2sk(xu); calculate y1(xu)*5Q1[z1(xu)*], Q1 being the normal score transform of the primary variable Z1(x);
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generate a value p from a uniform distribution U(0,1); generate a value ys from G[ y1(xu)*,s2sk(xu)]: ys5G21[y1(xu)*,s2sk(xu),p]; return the simulated value z1s(xu)5Q121 (ys) of the primary variable.
3. Loop until all nodes are simulated. The same DSS algorithm is applied to simulate Z2(x) assuming the previously simulated Z1(x) as the secondary variable. Co-located simple co-kriging is used to calculate z2(x)* and s2sk(xu), local mean and variance of Z2(x), conditioned to neighbourhood data z2(xa) and the co-located datum z1(xu) (Goovaerts 1997): z2 ðxu Þsck ~
n X
la ðxu Þ½z2 ðxa Þ{m2 zlb ðxu Þ½z1 ðxa Þ{m1 zm2
ð1Þ
a~1
Steps 2 and 3 are performed for the variable Z2(x). 4. 4.1
Experimental results Local co-regionalization model
Global Pearson correlation coefficients between hard and soft data (i.e. scaled posterior probabilities from the ML classification for the nine 30630 m grid cells centred at hard data locations) are 0.60 for eucalyptus, 0.50 for umbrella pine, 0.57 for maritime pine, 0.74 for cork oak and 0.55 for other cover classes. However, as usual in these situations, the global spatial correlation cannot be considered homogeneous and representative of the entire image (see figure 2): there are some local areas with high and low correlation coefficients for all cover classes. Hence, a model that reproduces those heterogeneities of the hard and soft data relationship is fundamental to ensure the quality of the classification. A local model of local coregionalization is applied, using the Markov-type approximation (Pereira et al. 2000) for CODSS, in order to accomplish the soft/hard data cross correlogram being equal to the product of local correlation coefficients with the correlogram of hard data. Local correlation coefficients between both datasets are calculated (within a given radius; Isaaks and Srisvastava, 1989), and adopted as the co-regionalization model of both variables in the CODSS procedure. The methodology succeeds in matching the global values of correlation coefficients, as well as the local correlation coefficients between the different types of data for all thematic classes.
Figure 2. Correlation surfaces for cork oak obtained by ordinary kriging of local correlations.
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For each of the 70 hard data points, local correlations between hard data, within a given radius, and the corresponding soft data were computed. Several neighbourhood radii were tested; a radius of 15,000 m was considered most appropriate considering both the number of neighbourhood samples involved and spatial correlation variability in the data. The local correlations were subsequently interpolated through ordinary kriging (Goovaerts 1997) for the whole study area, on a 90 m690 m grid. This way, correlation surfaces were obtained for each of the forest species and for the sum of the remaining coverage classes (figure 2). 4.2
Coverage proportion estimates
The simulation of the histograms for the field data measurements was hampered by a problem: their shape. These histograms are bi-modal with a high spike (comprising more than 70% of values) at the origin (see example in figure 3), which CODSS is unable to reproduce. Hence, we decided to apply CODSS not on the original data but on their cumulative probability—a more continuous and uniformly distributed variable. This procedure has the advantage of allowing us to back-transform the resulting simulations to their original distributions, thus reproducing the histogram of the field data. Prior to the CODSS procedure, hard and soft data were transformed to a uniform distribution U(0,1) with mean50.5. Ten simulations were computed for each of the forest species and for the sum of other land covers, using co-located co-kriging estimates (Xu et al. 1992, Almeida and Journel 1994). The statistics of each simulation result were compared to those of the hard data. As previously stated, all simulations are equiprobable and the mean of these 10 simulations is the most likely scenario. Hence, all simulations were averaged and back-transformed to the original scale. Before the back transformation, all simulations had approximately uniform distributions with statistics similar to those of the transformed data. Inherent to the procedure, simulations respect the hard data at their locations. After backtransformation, each simulation reproduces approximately the histogram and variogram of the original hard data, while the simulation average has statistics similar to those of the original soft data. An example is given for cork oak (figure 4), and analogous results were obtained for all cover classes. The mean simulated map obtained for all the cork oak through CODSS, and the simulation variance is presented in figure 4 (left column). In comparison to the coverage probabilities derived from the classification of the remotely sensed data (see figure 1, left column), which were used as soft data input, the mean of the simulations shows smoother coverage proportion distributions. Forest species distribution does not always match, e.g. umbrella pine appears in the upper western corner of the study area in the simulation, but not in the satellite image classification. In order to obtain a measure of uncertainty, simulation variance and interquartile range were computed for each grid cell, using the set of 10 simulated images. These measures of local uncertainty are clearly one of the biggest advantages of the proposed approach. The difference in the results for the study area, between these two methods of local uncertainty evaluation, resides in a scale difference, not showing spatial discrepancies. Spatial uncertainty measured by the variance of the simulated images (see right column of figure 4) is closely related to local correlation. Comparison of the variance
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Figure 3. Example of CODSS applied to cork oak forest cover: (a) transformed soft data; (b) variogram model for the transformed hard data (black curve) and experimental variogram for one of the simulations (thicker grey curve); (c) statistics of the hard data and (d) statistics of one back-transformed simulation (both with cork oak cover percentage on the x–axes).
and the local correlation surfaces (figure 2) reveals coincidence between low correlation and high uncertainty zones, and vice versa. Furthermore, uncertainty appears to be less for areas with higher and more homogeneous proportions of land covered by the respective class, whereas more uncertainty is present in the boundaries between two coverage classes. The spatial uncertainty can be used to produce maps identifying and/or classifying areas that need to be re-sampled and monitored for a better planning of those forest resources (Carvalho 2002). In figure 5, an example is given, applying a threshold of half the simulation variance. 4.3
Forest coverage estimates
The methodology applied in the previous section provided us with a coverprobability map for each cover class—eucalyptus, umbrella pine, maritime pine, cork oak and other covers—through CODSS. In order to produce a multiphasic map that combines all of these covers into one map, a dynamic classification procedure was used (Soares 1992, Goovaerts 1997). This algorithm is based on two criteria: (1) the maximization of the local probability of each grid cell, i.e. each grid
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Figure 4. CODSS coverage proportion estimates (left) and estimation variance (right) for the cork oak.
cell is assigned to the most likely cover class; (2) constraining classification to the reproduction of global proportions for the different classes, i.e. the grid cells with the highest probabilities for a given class are assigned to that class until the class’s global proportion is reached. The classes’ cover proportions for the study area given by the original classified satellite image were taken as indicative of the classes’ global proportions.
Figure 5. Areas needing resampling (grey): threshold applied to half the variance; and sampled areas (point).
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Analogous to the cover-proportion maps (figure 4), the dynamic classification yields a smoother image than the ML a priori classification (for comparison, upscaled to the 90690 m resolution used for simulation; figure 6). Classifications differ most for the umbrella pine (with only 21% overlap) and maritime pine cover (26% overlap), with large areas originally classified as maritime pine and ‘‘others’’, respectively (table 2). Notice that the distinction between umbrella and maritime pine cover was particularly difficult in the classical classification (cf. table 3). Comparison of the original 30630 m ML classification is slightly more divergent, as it is evidently less smooth. 4.4
Comparison between geostatistical method and classical classification
To facilitate visual comparison of the two classification results, a separate map of classification differences is presented for each of the coverage classes (figure 7). Differences comprise patches of apparent cover underestimation of the ML classification for both pine species. Differences in cork oak classification mainly occur in the vicinity of areas dominated by this species’ cover, suggesting the smoothing character of simulation in comparison to standard classification. The image of differences related to the sum of other classes appears to mirror the
Figure 6. Forest cover classification obtained from the upscaled classified satellite sensor data (left) and from the CODSS (right).
Table 2. Comparison of land cover percentages of the study area according to the ML classification and the CODSS. ML Classification
CODSS classification
Eucalyptus Umbrella pine Maritime pine Cork oak Other
Eucalyptus
Umbrella pine
Maritime pine
Cork oak
Other
41.6 8.2
3.3 21.1
5.2 49.4
16.3 2.4
33.5 18.9
2.8
6.3
26.1
4.6
60.2
4.4 3.2
0.4 1.6
0.8 2.9
62.7 5.6
31.7 86.7
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Table 3. Percentage of correctly (in bold) and miss-classified training areas, for the maximum likelihood and CODSS classifications. Training area cover class Eucalyptus
Umbrella pine
Maritime pine
Cork oak
Other
ML Eucalyptus classification Umbrella pine Maritime pine Cork oak Other
87.0 2.6 7.2 1.8 1.5
3.7 52.3 30.3 0.8 13.0
20.6 9.9 55.9 8.8 4.9
0.6 0.3 1.1 92.2 5.9
0.4 2.6 1.7 6.0 89.4
CODSS Eucalyptus classification Umbrella pine Maritime pine Cork oak Other
82.5 8.7 2.5 0.0 6.2
2.1 59.5 1.9 0.0 36.5
21.7 5.8 39.8 8.4 24.3
0.0 0.0 0.0 99.0 1.0
3.5 0.6 15.4 3.2 77.2
assemblage of the forest–species’ images. Classification differences are correlated to species cover proportions, as differences occur mainly in regions where the cover has a high proportion, but show no positive correlation with local variances. Cross-validation was performed on the 156 training areas (used for the ML classification) within the study area that were not used as hard data for the geostatistical calibration procedure. CODSS calibration predicts eucalyptus, cork oak and other coverage well (.75% of correctly predicted grid cells). Overall the geostatistical approach classifies 75% of the training areas correctly. Taking into account that for ML classification this is not a cross validation and the results, in all rigor, cannot be compared, classical classification appears to perform better than the geostatistical methodology for eucalyptus, maritime pine and the other cover classes, whereas CODSS performs better for umbrella pine and cork oak (table 3). There are differences in classes attributed to the miss-classified training data; CODSS attributes more data unduly to the class of other covers (e.g. for the pine
Figure 7. Difference between the CODSS and satellite sensor data classification (MLC); white: CODSS5MLC; grey: CODSS yields pictured class and MLC does not; black: MLC yields pictured class and CODSS does not.
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species) than ML classification. Overall, classical classification correctly predicts more when compared with the training areas. This is not surprising as remotely sensed data classification was assisted by these training data. However, when comparing the results of the two classification procedures with the contemporary inventory, performances are inverted: the classical approach of classification presents a difference of 8% in comparison to the land surface coverage inventory, while the geostatistical calibration presents a difference of only 3%. In terms of each forest species, both procedures underestimate the cork oak and eucalyptus, and overestimate maritime and umbrella pine. Overall, the classical approach leads to an over-classification, whereas the geostatistical method results in an under-classification of the forest species. 5.
Discussion and conclusions
This study proposes geostatistical satellite sensor data classification based on the stochastic CODSS procedure and on local co-regionalization models. The applied simulation technique provides a closer-to-reality image of forest cover, and enabled estimations of forest cover probabilities for the Setu´bal Peninsula, as well as local uncertainty assessment. In comparison to the classical classification, the geostatistical calibration yielded smoother distributions of forest species cover probabilities. ML classification failed to reproduce the coverage observed in many of the training locations used for its calibration and in many of the field observations we used as hard data. The latter were inevitably perfectly reproduced by the simulations, which were conditioned to them. In the cross-validation on the training areas previously used for the ML classification, CODSS and ML classification performed differently for the different cover classes. CODSS performed better than classical classification for eucalyptus and cork oak and worse for umbrella and maritime pine. Overall, both methods failed to classify numerous training areas of both pine species correctly. On the other hand, most eucalyptus and cork oak training areas were correctly classified, probably because these two forest resources have more continuous distributions on the Setu´bal Peninsula, unlike umbrella and maritime pine, which are mainly scattered and intermingled. Limitations to this study were posed by the quality of the collected hard data, as sampling was not stratified over all classes and therefore not representative of the studied area. Furthermore, the aggregation of the original satellite sensed data classification map to the lower study spatial resolution implies a loss of information, with penalization of less frequent and more scattered species. During upscaling, each new grid cell was assigned to the most frequent class of the nine underlying 30630 m grid cells, penalizing less frequent cover classes. This may partly explain the simulation methods’ failure to improve on eucalyptus and maritime pine classification—species that frequently cover small areas. Upscaling of the posterior probabilities for these grid cells might have been a more correct alternative. Another constraint is that simulations are based on the remotely sensed data classification as soft data, which is not always optimal. During the two years between the collection of the training data, used for ML classification and for the validation of both classification methods, and the collection of some of the field observations used as hard data in the CODSS, some land cover may have changed, accounting for part of the misclassified training areas.
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Future research should focus on improvements in the soft and hard data. In this context, the local uncertainty assessment provided by the simulation approach will allow the identification and classification of areas that need to be re-sampled and monitored for better management of those forest resources. Finally, validation should be performed on an independent dataset. References ALMEIDA, A. and JOURNEL, A.G., 1994, Joint simulation of multiple variables with a Markovtype co-regionalization model. Mathematical Geology, 26, pp. 565–588. ATKINSON, P.M. and LEWIS, P., 2000, Geostatistical classification for remote sensing: an introduction. Computers and Geosciences, 26, pp. 361–371. CARVALHO, J., 2002, Stochastic simulation for satellite images calibration. Masters thesis, Instituto Superior Te´cnico. DELGADO, J., SOARES, A. and CARVALHO, J., 2004, Landsat-SPOT digital images integration using geostatistical cosimulation techniques. In ISPRS 2004—XXth Congress of the International Society for Photogrammetry and Remote Sensing, 12–23 July 2004, Istanbul, Turkey, Vol. XXXV (B4), pp. 895–900. GOOVAERTS, P., 1997, Geostatistics for Natural Resources Characterization (New York: Oxford University Press). GOOVAERTS, P., 2000, Geostatistics approaches for incorporating elevation into the spatial interpolation of rainfall. Journal of Hydrology, 228, pp. 113–129. ISAAKS, E. and SRISVASTAVA, R.M., 1989, An Introduction to Applied Geostatistics (New York: Oxford University Press). LILLESAND, T.M. and KIEFER, R.W., 1994, Remote Sensing and Image Interpretation, 3rd edn (New York: John Wiley & Sons). PEREIRA, M.J., 1998, Air quality modelling and simulation. PhD thesis, Instituto Superior Te´cnico. PEREIRA, M.J., SOARES, A. and ROSA´RIO, L., 2000, Characterization of forest resources with satellite SPOT images by using local models of co-regionalization. In Geostatistics 2000 Cape Town, edited by W.J. Kleingeld, and D.G. Krige (Eds), Vol. 1, pp. 581–590. PROJECT LIFE ENV/P/00556 2002, The atmospheric pollution and the management and conservation of forest ecosystems in the Setu´bal peninsula: Final report, Association of Setu´bals Peninsula Forest Producers, AFLOPS, Portugal. SCHOWENGERDT, R.A., 1997, Remote Sensing, Models and Methods for Image Processing, 2nd edn (San Diego, CA: Academic Press). SOARES, A., 1992, Geostatistical estimation of multi-phase structures. Mathematical Geology, 24, pp. 149–160. SOARES, A., 2001, Direct sequential simulation and cosimulation. Mathematical Geology, 33, pp. 911–926. SOARES, A., PEREIRA, M.J., BRANQUINHO, C. and CATARINO, F., 1997, Stochastic simulation of lichen biodiversity using soft information from remote sensing data. In GeoENV I—Geostatistics for Environmental Applications, A. Soares, J. Go´mez-Herna´ndez, and R. Froidevaux (Eds) (Dordrecht: Kluwer Academic), pp. 375–387. XU, W., TRAN, T., SRIVASTAVA, M. and JOURNEL, A.G., 1992, Integrating seismic data in reservoir modelling: the collocated co-kringing alternative, SPE paper 247242.
