Stochastic Modeling of Continuous Geospatial or Temporal Properties

Abstract

This chapter presents geostatistical methods for stochastically simulating continuous geospatial properties. For facilitating the presentations, it uses many temporal data in stochastic simulations benefiting from the 1D simplification. The commonly used simulation methods for spatial data include sequential Gaussian simulation and spectral simulation. Unlike estimation methods (e.g., regression and kriging), one main goal of stochastic simulation is to model the heterogeneities of physical properties.

Stochastic simulations are often mathematically extended from estimation methods. Therefore, the kriging methods presented in Chap. 16 are used as a methodological basis for stochastic simulations. Readers should be familiar with kriging, especially simple kriging, before reading this chapter. The main texts in this chapter focus on basic methodologies and three appendices cover more advanced topics.

The true logic of this world is the calculus of probabilities.

James Clerk Maxwell

A model is just an imitation of the real thing.

Anonymous

Appendices

Appendices

1.1 Appendix 17.1: Ergodicity, Variogram, and Micro-ergodicity

Ergodicity plays a critical role for statistical inference because it deals with the problem of determining the statistics of a stochastic process for a single realization. The notion of ergodicity originated from statistical mechanics, when Gibbs observed that in a closed system where the total energy remains constant, a time average over the motions of a system of particles has the same average obtained by integration over a surface in phase space, called the ergodic surface (Lee 1967; Lebowitz and Penrose 1973). Because the word “ergodic” means “working path”, ergodicity opens up the path for statistical inference of stochastic processes with a single realization. One straightforward way of thinking about the classical ergodic theorem is that it is a generalization of the Law of Large Numbers (Chap. 2) because it implies that a sufficiently large sample size is representative of the population.

Many conventional stochastic methods assume the ergodicity (Papoulis 1965; Lee 1967; Gray 2009), as Matheron (1989, p. 81) stated “From the classical point of view, the possibility of ‘statistical inference’ is always, in the final instance, based on some ergodic property.” Zhan (1999) raised a concern of using the ergodicity assumption when the heterogeneity of the property is strong but concluded that it is valid when the variance of the property is not too high. Measures for checking the applicability of the ergodicity are given by Zhan (1999), and Helstrom (1991), including the (relative) magnitude of variance and covariance function.

An IRF-0 Z(x) does not have a constant variance, but the variance of its first-order difference depends only on the lag, not the spatial location of the RF Z(x). The variance of the first-order difference was initially termed the serial variation function (Matern 1960; Pettitt and McBratney 1993) and was later termed a variogram or a semivariogram (Journel and Huijbregts 1978; Matheron 1989). One advantage of the variogram is that for short lags, it is essentially independent of long-term variation in the series and requires no reference to the series mean, a quality that makes it suitable in the study of local variation (Pettitt and McBratney 1993). This advantage is especially distinct when comparing it to Fourier analysis, in which the spectra are calculated with the full range of the defined domain, and theoretically calculated from – ∞ to +∞. For limited fields, the Fourier transform sometimes produces less reliable spectra in describing the frequency content of phenomena (Ma 1992; incidentally, by using a local neighborhood, kriging can be used to estimate the spectrum for limited dataset, see Appendix 17.3). But the micro-ergodicity makes the variogram more robust in dealing with limited data.

Unlike the ergodicity in the conventional second-order statistics, the micro-ergodicity concept is used in the IRF framework. A statistical parameter is micro-ergodic if it is fully determined by a single realization of its RF on a limited field (Matheron 1989; Stein 1999). For instance, the variogram is micro-ergodic near the origin, i.e., for short lag distances, if it is not too regular (great regularity tends to represent a deterministic function instead of a stochastic field). In contrast, many traditional statistical parameters are not micro-ergodic on a limited field. Physically, the micro-ergodicity emphasizes the neighboring resemblance and contextual information.

Micro-ergodicity links the characteristics of the variogram with the regularity of random fields. A discontinuous variogram at the origin implies that the random field is a white noise or contains a white noise component (Fig. 17.13a). A continuous variogram with a linear property at the origin implies the continuity of the random field in the mean-squares sense (Fig. 17.13b). A variogram derivable at the origin implies that the RF is derivable in the mean-squares sense (Fig. 17.13c).

Fig. 17.13

Three different behaviors of a variogram at short lag distances. The horizontal lines represent the variance

The concept of micro-ergodicity forms a foundation to use local operators in stochastic modeling. Many nonstationary processes can be considered as locally stationary (Papoulis 1965; Matheron 1989; Ma et al. 2008), and thus can be dealt with using simpler modeling methods. This concept enables emphasizing the neighborhood dependency. Only in some special situations, are the IRF-k theory and universal kriging method more effective in practice. Moreover, the micro-ergodicity concept is also applicable to the spatial correlation or covariance function. Therefore, the covariance can be used in the kriging system instead of the variogram.

Two end members are discontinuity at the short lag distance and continuity to the degree of being derivable for a variogram. These are the cases of the (partial) nugget effect variogram and Gaussian variogram. The nugget effect variogram is discontinuous at the zero-lag distance because of the presence of white noise. The Gaussian variogram represents a strong continuity, implying a very smooth RF, derivable in the mean-squares sense. Exponential and spherical variograms are continuous at the zero-lag distance.

1.2 Appendix 17.2: Spectral Representations of Variogram and Covariance Functions

There are advantages to representing a stochastic process, its variogram and spatial covariance function in the frequency domain. In the context of generating a stochastic realization of a reservoir property using spectral method, the relationship between the variogram and covariance function (Eq. 13.5 in Chap. 13) is used because the relationship between covariance function and spectral representation is well studied in the literature. Limited studies have been carried out for direct analysis on the variogram and spectral relationship. The four exponential variograms shown in Fig. 13.3b of Chap. 13 can be first converted into covariance functions, and then Fourier transforms of those covariance functions will give their spectra (Fig. 17.14).

Fig. 17.14

Frequency-spectrum plots for the four exponential variograms in Fig. 13.3b (Chap. 13; after converting them into covariance functions). The additional flat line is the spectrum of a pure nugget effect. Note that the amplitude spectra for the models with a = 60 and a = 30 were partially truncated in the displays because the spectral values for the low frequencies are beyond the scale of the figure

Clearly, the smaller the correlation range, the higher the proportion of higher-frequency content. The pure nugget effect can be considered to have zero correlation range; its spectrum is a flat line, implying a lot of high frequency. Some may wonder how a flat line can be interpreted as containing a lot of high frequency content. In fact, most natural phenomena have a significant amount of low-frequency content and little high-frequency content, contrary to the common perception. When the high-frequency content is as much as the low-frequency content, it is a significant amount of high frequency. This is the case of a white noise or pure nugget effect.

The spectra of most covariance functions (positive definite by definition) can be analytically derived. An exponential covariance function and its spectrum are expressed as follows:

$$ C(h)=\exp \left(\hbox{--} ah\right) $$

(17.16)

$$ S(f)=2a/\left({a}^2+4{\uppi}^2{f}^2\right) $$

(17.17)

A hole-effect variogram based on multiplication of exponential and cosine functions and its spectrum are:

$$ C(h)=\exp \left(\hbox{--} ah\right)\cos \left( 2\uppi h\right) $$

(17.18)

$$ S(f)=\frac{a}{a^2+{\left(2\pi f-2\pi \right)}^2}+\frac{a}{a^2+{\left(2\pi f+2\pi \right)}^2} $$

(17.19)

where h is the lag distance, and a is the decay parameter.

Another advantage of using spectral simulation is the ease of defining a positive definite covariance function. The Bochner theorem states that a positive spectrum is a necessary and sufficient condition for its covariance function to be positive definite (Matheron 1988). In this regard, it is not even necessary to fit a variogram model. It is possible to calculate an experimental variogram from the data and then use the Fourier transform to convert it into frequency domain. Following the Bochner’s theorem, one can simply set all the negative spectral values to zero; and the spectrum will represent a positive definite function. Figure 17.15 shows an example of experimental correlation function and its spectral representation in the frequency domain; a few very small negative values were set to zero.

Fig. 17.15

(a) Experimental spatial correlation of a geospatial variable. X axis is the lag distance in meter. Y-axis is the correlation. (b) Spectrum of (a). X-axis is frequency, cycle per meter. Y-axis is the amplitude spectrum

Because of the relationship between a variogram and covariance function, as the equivalency of the covariance function being positive definite, the variogram must be conditionally negative definite; see Lantuejoul (2002) for more detail.

1.3 Appendix 17.3: Estimating Spectrum from Limited Data Using Kriging: 1D Example

A time series or spatial variable can have a spectral representation. The spectrum can be estimated using maximum entropy and ARMA (autoregressive and moving average) methods (Marple 1982; Fournier and Ma 1988). Simple kriging can also be used to estimate spectrum (Ma 1992).

Although estimating spectrum using kriging has not been commonly practiced, it shows how kriging is related to methods of time series analysis and stochastic simulation using spectrum. We briefly review this method here.

For the purpose of demonstration, consider regularly sample 1D data, in which we estimate a value at location x with a symmetrical window of n data points each side. When the value to be estimated is part of the known data, simple kriging will be equal to that value because of its exact interpolator property. Therefore, the linear combination of the estimator by excluding that datum is

$$ {Y}^{\ast }(x)={\sum}_{j=-n}^{j=n}{w}_j\mathrm{Y}\left({x}_j\right)\kern1em \mathrm{for}\ \mathrm{j}\ne 0 $$

(17.20)

where w _j are weights and Y(x _j) are the data. This is very much the same as a regular kriging estimator, except using a symmetrical window in a regular sampled dataset.

The estimation error is

$$ e(x)=Y(x)-{Y}^{\ast }(x)=Y(x)-{\sum}_{j=-n}^{j=n}{w}_j\mathrm{Y}\left({x}_j\right)\kern1em \mathrm{for}\ \mathrm{j}\ne 0 $$

(17.21)

Applying the Z transform to Eq. 17.21 leads to

$$Y\left(\mathit{Z}\right)=\frac{\mathit{e}\left(\mathit{Z}\right)}{1-{\sum }_{\mathit{j}=-\mathit{n}}^{\mathit{j}=\mathit{n}}{\mathit{w}}_{\mathit{j}}\mathrm{Y}\left({\mathit{x}}_{\mathit{j}}\right)}\text{for}\mathrm{j}\ne 0$$

(17.22)

Setting

$$ \mathrm{z}=\exp \left(-2\uppi \mathrm{if}\Delta \mathrm{x}\right) $$

(17.23)

where i is the complex number, f is the frequency, and Δx is the temporal or spatial step (or lag). Calculating the square of Eq. 17.22 leads to the power spectrum of Y(x):

$$S\left(\mathit{f}\right)=\frac{{\sigma }_{\mathit{x}}^2}{{\left|1-{\sum }_{\mathit{j}=-\mathit{n}}^{\mathit{j}=\mathit{n}}{\mathit{w}}_{\mathit{j}}exp\left(-2\mathrm{\pi }\text{if}\mathrm{\Delta }\mathrm{x}\right)\right|}^2}\text{for}\mathrm{j}\ne 0$$

(17.24)

where ${\sigma}_x^2$is the kriging estimation variance, and the frequency f is limited to the Nyquist interval.

As a result of the configuration symmetry, the kriging weights are symmetrical. Thus Eq. 17.24 can be simplified to

$$S\left(\mathit{f}\right)=\frac{{\sigma }_{\mathit{x}}^2}{{\left|1-{\sum }_{\mathit{j}=1}^{\mathit{j}=\mathit{n}}{\mathit{w}}_{\mathit{j}}\left[exp\left(-2\mathrm{\pi }\mathrm{ifj}\mathrm{\Delta }\mathrm{x}\right)+exp\left(-2\mathrm{\pi }\mathrm{ifj}\mathrm{\Delta }\mathrm{x}\right)\right]\right|}^2}\text{for}\mathrm{j}\ne 0$$

(17.25)

Applying the Euler formula to Eq. 17.25 leads to

$$S\left(\mathit{f}\right)=\frac{{\sigma }_{\mathit{x}}^2}{{\left|1-2{\sum }_{\mathit{j}=-\mathit{n}}^{\mathit{j}=\mathit{n}}{\mathit{w}}_{\mathit{j}}cos\left(2\mathrm{\pi }\mathrm{fj}\mathrm{\Delta }\mathrm{x}\right)\right|}^2}\text{for}\mathrm{j}\ne 0$$

(17.26)

Unlike estimating an unknown value where the initial linear combination is used to obtain the final estimation, the power spectrum is a cosine transform of the kriging weights. Figure 17.16 shows two examples of estimated spectra of short window mixed sinusoids using simple kriging. The comparison with the autoregressive method can be found in Ma (1992).

Fig. 17.16

Estimated spectra of sinusoid(s) by simple kriging: the left figure is for a 50 Hz single sinusoid and the right figure is for a mixed signal with a 25 Hz sinusoid and a 50 Hz sinusoid. Both signal are sampled in a short window