Geostatistics is the analysis of spatially continuous data. It treats geographic attributes as mathematical variables depending on locations. One of the central techniques among geostatistical approaches is the variogram, which describes to what extent spatial dependence exists between sample locations. It can be used in spatial interpolation with empirical models, which has particular applications such as oil exploration. Geostatistics in remote sensing (RS) has thrived in the last several decades (Atkinson & Lewis 2000). Multispectral data is unique, due to the "continuous" characteristics across several waveband layers. Spatial autocorrelation, introduced by target feature and sensor properties, allows examination of geospatial techniques involving image processing. In this review paper, I will focus on the application of variogram in multiband image classification.

1. Variogram in Remote Sensing

Geostatistical techniques evaluate autocorrelation commonly observed in spatial data (Wallace et al. 2000). The spatial autocorrelation structure can be depicted by the variogram, which is the semivariance plotted against distance under the intrinsic hypothesis (Bailey & Gatrell 1995). The experimental variogram is calculated as one-half of the observed average square difference in data values for every pair of data locations at a separation distance (h), which can be expressed as:
(1)
where is a data location, is a lag vector, is the data value as the location . is the number of data pairs at distance . The so-derived variogram is then used to fit an appropriate theoretical model variogram of known mathematical properties, which allows values of unsampled locations to be estimated based on the input variogram (Wallace et al. 2000).

The parameters of the model variogram include the sill, range and nugget (Bailey & Gatrell 1995). Sill is defined as the total variation present. Range is the distance over which data are correlated. Nugget represents the extent of random variation within the data. According to equation (1), variance at the distance of 0 is theoretically equal to 0. However, sampling errors and small scale variability may often cause sample values within a small separation to be quite dissimilar. Thus, a discontinuity at the origin of variogram occurs, which is referred to the nugget effect.

In the context of remote sensing, images can be treated as "field" data depicted by varied digital numbers (DN). The spectral values of pixels are commonly spatially autocorrelated. Therefore, image texture based on spatial autocorrelation might be efficient to improve RS classification, by integrating spatial information of the original radiometric data (Chica-Olmo & Abarca-Hernández 2000). An important issue stems is the distinction between local variability and spatial autocorrelation in terms of texture. The local variability is defined as the variance (), which is a statistical measure of DN dispersion with respect to the mean value of a moving window (Woodcock & Harward 1992). The spatial autocorrelation is hence treated as spatially dependent structure (rather than randomly distributed) associated with a given land-cover class within the moving window (Lark 1996).

Under the intrinsic hypothesis, variogram describing the spatial autocorrelation of the radiometric data can be expressed as
(2)
where is the number of pixel pairs at a certain lag . is the digital number at the location . Thus, represents half of the observed average differences of the quadratic increment/decrement of paired pixel values at a distance (Atkinson et al. 2000, Chica-Olmo & Abarca-Hernández 2000). For a given , the corresponding semivariance values can be calculated within the moving window, which is defined according to DN variance regarding land-cover types. Consequently, the so-derived layer of semivariance values can be incorporated to RS classification as an extra "texture channel".

As Chica-Olmo & Abarca-Hernández (2000) remarked, the spectral autocorrelation of radiometric data is applied based on variographic analysis. Computing the variogram function does not present any difficulty theoretically, given the raster structure of a satellite image. Modeling is not absolutely necessary, because only experimental values of the semivariance are used. However, the parameters of sill, range, nugget and spatial anisotropy are still important in the procedure of determining the sizes of moving window and lag distance to be used to compute the essential semivariance.

2. Variogram Training and Semivariance Computing

Miranda et al. (1998) reported that distinctive spatial autocorrelation patterns exist in RS images, depending upon different land-cover types. In their study, the authors created a Semivariogram Texture Classifier (STL), and applied it to the classification of JERS-1 (Fuyo-1) SAR data in a rainforest of the Uaupés River (Brazil). The potential land-cover types under investigation included upland dense vegetation (characterized by diffuse radar backscatter), open vegetation in dry depressed area (where single forward scattering predominates over diffuse backscatter), flooded vegetation (where standing water beneath the canopy causes double forward reflection of microwave energy in L-band), and water (characterized by specular reflection of the incident radar beam).

An arbitrary size of training sites (22×22 pixels) was determined for computing variograms of every land-cover type. The variance of DN within a moving window was 182 for water, 779 for open vegetation, 1474 for upland dense vegetation, and 2527 for flooded vegetation. The unique variogram behavior was found associated with each corresponding vegetation type (Fig 1). Variogram of water was essentially flat, exhibiting little or no spatial correlation for lag distances further than 1 pixel. Variogram of open vegetation rose up to a lag distance of 2 pixels and then curved flat, fairly coincided with the DN semivariance of the training site. Variogram of upland dense vegetation reached sill at a lag of 2 pixels. The sill value was larger than that of the open vegetation class, which indicated that its DN values displayed a greater variance than open vegetation. Variogram of flooded vegetation showed the greatest variance and rose steadily toward the sill at lag distances further than 15 pixels.

The study by Miranda et al. (1998) showed a good example of how variogram can be used to represent typical land-cover features, based on the spatial information that is inherently contained in multi-spectral dataset. Given a matrix of pixels (× window) from the identical class, the semivariances corresponding to a certain lag may be unique for different land-cover types. Therefore, this kind of information can be recoded to generate an ancillary data layer, which contains semivariance value for each pixel calculated at the pre-determined lag distance within the pre-defined moving window. The determination of both the window size and the lag size appears to be the core procedure to derive meaningful variogram curves. The procedure of finding out these parameters is referred as variogram training.

In terms of the window size, obviously it should not be too large, as compared with the average land feature sizes (Chica-Olmo & Abarca-Hernández 2000). If the moving window is larger than the patch of variogram training site (which is associated with particular land-cover type), the variogram curve will be biased due to the presence of non-typical pixels from other land-cover types. Thus the resulted semivariance value based on the certain lag will be biased, because the determined lag itself is no more representative. Meanwhile, the moving window should not be too small either (Chica-Olmo & Abarca-Hernández 2000). If it is too small, all one can get may be pure nugget, which means no spatial autocorrelation (therefore, no texture exists). In addition, the available lag distance is limited by the extension of the moving window. If the moving window is an × matrix, the possible lag size is only up to .

To determine the lag size, variogram should be computed for each of the known land-cover type at the training site. All of the variogram curves should be plotted against the lag distance sequence on a single sheet, so as to compare the variogram patterns simultaneously. The lag size is determined by reflecting both the spatial autocorrelation characteristics in raw image data and the unique semivariance values linked with different land-cover types. For example, the optimal lag size might be 2 pixels, according to the variogram patterns in Fig 1.

After determining the moving window size and the lag size, the semivariance can be calculated at each pixel by shifting the moving window throughout the entire image. Practically, the semivariance values are assigned to the central pixel of the moving window. The raster "texture channel", which can be added to the original spectral dataset to facilitate more accurate classification, is formed essentially.

3. Modified Variogram and Classification Efficiency

The method reviewed in the first and second parts can be identified as the "uni-viriogram". Such variogram is calculated directly from equation (2), usually applied to a single radiometric layer of image (such as the NIR band of the ETM+ image). Modified versions based on this basic equation are also available in RS classification. They have proven quite efficient in improving the classification accuracy, according to Chica-Olmo & Abarca-Hernández (2000). In their study, they compute four kinds of variogram that include:

• The direct variogram based on equation (2)
• Madogram, which is similar to the direct variogram, but uses the absolute differences instead of the squared differences. The corresponding equation can be expressed as

(3)
• Cross variogram, which quantifies the joint spatial variability between two bands. The equation can be expressed as (4)

where and are the radiometric bands.
• Pseudo-cross variogram, which represents the semivariance of the cross increments instead of the covariance of the direct increments as in equation (4). The corresponding equation can be expressed as

(5)

Fig 2. Direct and Cross Variograms. (dotted lines: alluvial terrace; dashed lines: undifferentiated Quaternary; and solid lines: piedmont)
(Source: Chica-Olmo & Abarca-Hernández, 2000)

The authors calculated "texture channels" for a Landsat-5 TM scene (550×350 pixels, 6 bands) in order to compare classification accuracy for classifying different kinds of sediments. A principle component analysis (PCA) was run beforehand to derive two major component channels, which were designated as the PC1 for visible bands and the PC2 for infrared bands. The direct and cross variograms within training sites of the three Quaternary deposits are shown in Fig 2. Accordingly, the lag size of 1 pixel in a moving window of 7×7 matrix was adopted. Anisotropy was also considered by taking into account of four main directions (N-S, E-W, N45E, and N45W).

Supervised classification was performed for both original radiometric image and image set including different texture content. The authors found that classification accuracy can be improved up to 20% when textural information derived from both PC uni- and cross-variogram is integrated with the raw image of 6 spectral bands. The cross variogram techniques were found more efficient (11.6%) in improving classification accuracy than the uni-variogram (9.7%) ones. The fractal dimension (considering anisotropy) alone was not able to improve accuracy very much (2.9%); however, it played an important role when combined with the uni- and/or cross- variance channels.

4. Summary and Discussion

The geostatistical technique of variogram has proven helpful in RS land-cover classification, according to proceeding studies discussed above. The texture information derived from land feature spatial autocorrelation may be code to create texture channels associated with the original radiometric data. These texture layers are composed of semivariance calculated within "blocks" of neighboring pixels at a specific lag distance. Therefore, the determination of "block size" (i.e., the size of moving window) and lag size is particularly important. These parameters are decided through experimental trainings for each of the known typical land-cover features. Basically, they are resulted from the variogram patterns involving sill, range and nugget effect. Several issues under emphasis in the reviewed papers include:

(1) Edge effect

For the calculation of semivariance at a certain lag distance within the × moving window, edge effect was not considered in either of the studies presented above. However, edge effect does exist, because pixels at the moving window boundaries do not have the same number of paired pixels at the separation of lag . Moreover, pixels might have lost some meaningful counterpart pixels, which are actually belonging to the same land-cover patch but be cut off by the moving window boundaries.

(2) In terms of texture

All of the authors have reported that different land-cover types usually have different variogram patterns, which are depicted by the absolute variance value (involving the sill) and the curve shape (involving with nugget and lag). Thus, to get a meaningful lag size is the function of both the degree of spatial autocorrelation and the separability of different features at the specific lag. For example, Miranda et al. (1998) determined the lag size as 3 pixels.

One of the problematic issues might be spatial autocorrelation to be interpreted at different scales, precisely the classification level. For example, the autocorrelation texture might be distinctive for forest, agriculture and urban settlement on an image scene. However, it might not be distinguishable for coniferous and deciduous forest stands, because their variance (sill) and spatial pattern might be similar. Or reversely, if the texture variogram is distinct for the sublevel forests, the higher level class texture might not be as useful due to the strong within-class variations. Thus, it is crucial to decide how many and what kind of classes one may get from the radiometric data before the experimental variogram training. I.e., the pre-knowledge is really important.

(3) Image resolution and computing demands

The resolution of image under analysis should not be too coarse, compared with the land-cover features’ dimension. Otherwise, all one can get is the pure nugget. Fortunately, this situation is rare if the image is spectrally capable for land-cover classification. In the meantime, the resolution should not be too fine either. Otherwise, the computing demands will be extremely high due to the large amount of pixel pairs within the moving window. Finally, the physical image size is also a problem at times. A 1000×1000 image needs at least 4 times work as the 500×500 one, under the same condition of resolution, moving window and lag size. If the resolution is finer or the lag distance is shorter, the former will require much more CPU memory and physical space to store the computational results.

Literature Cited