Analyzing spartially continuous data

ArcGIS’s Geostatistical Analyst provides both deterministic and geostatistical (stochastic interpolation) methods to: (i) deduce the characteristics of the spatial or spatio-temporal variation in the values of a variable (e.g., fault trend) in the study area based on the sampled point values, (ii) model the distribution pattern of the values of a spatial variable, and reveal factors that might associate with it, and (iii) constructa continuous surface by interpolating and predicting the val- ues at unmeasured locations using the values of measured sample points (Saveliev et al., 2007; Esri, 2011). Geostatistical interpolation methods such as Kriging are based on statistics that op- timize prediction of unknown values by generating prediction surfaces and computing and as- sessing uncertainty surfaces toverify the accuracy of the predictions (Liu, 2003).

3.3.3.1 Kriging

Matheron (1960), a French mathematician and geologist, defined the concept of Kriging for predicting gold deposited in a rock from different core samples. Kriging estimates the value at unsampled locations by weighting the neighboring measured sample values. The neighboring measured sample values are weighted based on (i) the distance between them and the prediction location, and (ii) the spatial autocorrelation among them (Royle et al., 1981; Davis, 1986; Lam, 1983, Child, 2004; Azpurua and Ramos, 2010).

There are two methods ofKriging: (i) Ordinary Kriging that calculates and uses a local mean in the prediction, and (ii) Universal Kriging that estimates an overriding trend in the data (Kleinschmidt et al., 2000;Schuurmans et al., 2007). The Ordinary Kriging model is:

(Eqn. 3.10)

where Z(s) is the variable value (e.g., trend of a fault) that need to be estimated, µ is the mean, and ε(s) is the error caused by the spatial dependence (if it exists). Ordinary Kriging assumes the mean (µ) is constant but unknown and can be estimated locally from nearby locations.

Assuming that the random process ε(s) is fundamentally fixed, the estimates for Z(s0) (i.e., value

at the prediction location) are essentially weighted averages of the data: (Eqn. 3.11)

where: Z(si) is the value at the measured sample point at the ith location, λi is an unknown weight

for the measured sample point value at the ith location, s0 represents the prediction location, and

N is the number of measured sample point value (Royle et al., 1981; Oliver, 1990). In the Ordinary Kriging method, which is applied for data with a trend, the weight (λi) is a function of

the variogram model of the measured point values, the distance to the prediction location, and the spatial autocorolation among the measured values around the prediction location. The Kriging method computes the empirical semivariogram for a large data sets, fits a model, produces the matrices, and then constructs a surface (predication) with the z-values.

The semivariogram is a plot of the semivariance between sample values against distance. Since Kriging is based on Tobler's first law of geography (Tobler, 1979), the values of adjacent or nearby features (variables) are more likely to be similar compared to the values of distant fea- tures. The probability of truth or falseness for this assumption is tested by quantifying the spatial relationship (autocorrelation) in the weights (measured sample values) from the semivariograms.

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Kriging assumes that some of the spatial variation in the measured values may be due to random processes, and requires that autocorrelation be evaluated.

Variogram (2) as a measure of the spatial dependence of a spatial random field, Z(s), is the variance of the difference between the measured field values var(Z(si) - Z(sj)) at two loca-

tions i and j. The semivariogram () is defined as:

γ(si,sj) = ½ var(Z(si) - Z(sj)) (Eqn. 3.12)

where var is the variance, si and sj are the measured field values, and Z(si,sj) are variable values

to be estimated. As the distance between si and sj decreases the difference in their values will al-

so decrease and the values are more likely to be alike. Therefore, the semivariogram depicts the best-fit trend for the spatial autocorrelation of the measured sample points. The trend first rises up to certain distance (critical distance), and then it flattens out. The distance where the trend levels out is known as the range (Burrough, 1986; Royle, 1981; Oliver, 1990). Samples that are separated by distances that are less than the range are spatially autocorrelated; those that are spaced wider than the range are not. The value at which the semivariogram reaches the con- stant level is called the sill. The height of the jump of the semivariogram above the origin is called nugget, which may represent either the measurement error, microscale variation, or both

(Bohling, 2005).

Two types of directional components influence the predictions: global trends and direc- tional influences on the semivariogram/covariance (known as anisotropy). The global trend can be determined by an overriding process applying mathematical formula (e.g., a polynomial) that affects all measurements in a deterministic method. Anisotropy for a random process reveals autocorrelation as a function of direction. It represents the existence of directional differences in spatial dependence (autocorrelation).

A directional influence (anisotropy) affects the sampled point values and the trend of the semivariogram. The existence of anisotropy indicates that in certain directions, adjacent or closely-spaced features are likely to be more similar than the values of distant feature(Li et al., 2013).

The Geostatistical Wizard extension of ArcGIS’s Geostatistical Analyst was applied to

investigate the existence of any directional influence on the semivariagram using Ordinary Kriging. The orientation and the midpoint position of the Basin and Range and cross normal fault traces were used as input in this analysis. The exponentialsemivariogram model, variable lag sizes, and number of lags =12 were chosen as options for the geostatistical analysis.

ArcGIS automatically calculates the nugget, range and sill for the selected options. The anisotropy option was also checked to find the directional influence (anisotropy) on the

semivariogram representing the trend of the autocorrelation among the fault trace orientation da- ta in each domain. The directional influence (anisotropy) is depicted by a best-fitting ellipse with major and minor principal axes. The azimuth of the ellipse’s major axis with respect to the North (000o) is specified by the angle ‘φ’ (Tables 3.5 and 3.5). The anisotropy ellipses (depicting the directional influence), which were determined for each spatial domain, were placed in their cor- rect orientation using the azimuth of their major axes (φ), at the center of their corresponding domains on the map to detect the regional variation of the principal directions relative to the axis of the SRP.

The trend for the linear directional mean (using the angle ) for the fault set was also drawn as a reference and for comparison. The surface (prediction) map, which is constructed by the Geostatistical Analyst as output, interpolates and predicts the orientation of fault traces in areas along the SRP where they are missing (not sampled).

The accuracy of the predictions was evaluated by cross validation, which develops the trend and autocorrelation models by removing each data location one at a time and predicting the related data values. The cross validation was also applied to compare the predict-

ed and measured orientation of the fault data points in each domain.