Determination and use of indicator cross-

FAKTS-derived indicator variograms can also find application for the conditioning of another popular pixel-based method: plurigaussian simulations. The principle

behind the method involves the generation of two or more Gaussian fields and their truncation at a specified number of thresholds in order to attribute discrete values representing the categories (Le Loc’h & Galli 1997; Dowd et al. 2003; Xu et al.

2006; Armstrong et al. 2011). Unlike the SIS method, the plurigaussian method permits consideration of the juxtapositional tendencies of the modelled categories:

contact relations among the categories, which describe juxtapositional trends, can be specified by using pre-defined contact templates (lithotype rules in: Armstrong et al. 2011; Dowd et al. 2003) or user-defined contact matrices (dynamic contact matrix in Xu et al. 2006). The inputs to this method consist of the proportions of the chosen categorical variables, their experimental indicator auto- and cross-variograms, a model of their spatial relations and the Gaussian correlation coefficients. FAKTS can be used to obtain proportions as explained above, in case of sparse or no data (e.g unconditional simulations of fluvial architecture). In addition, instead of inputting experimental (cross-)variogram values, it is possible to derive indicator (cross-)variogram values from sampling at a given lag-spacing (cross-)variogram models generated using FAKTS data for each categorical variable (or each pair of categorical variables) (figure 6.10). FAKTS can be used to generate models of indicator auto-variograms for material units as explained above.

Also, since FAKTS stores data about the spatial relationships between genetic units, the database can be used to derive models of indicator cross-variograms for each pair of indicators. This would be done by exploiting the relationship between indicator-cross variograms of a pair of categories and their continuous-lag transition probability (Carle & Fogg 1996). The sill of the indicator cross-variogram model can be computed from category proportions, as it approaches –pjpk (Carle & Fogg 1996), where the j and k subscripts denote the two categories and p their proportion. As an approximation (downward) of the range of the model indicator cross-variogram for two categories, it is possible to assume the lag value at the intersection between the sill (pk) of the continuous-lag transition probability for the same categories and the tangent (s) to the same transition probability at lag zero.

The lag value at this intersection is given by:

x

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Notably, working with embedded categories means considering self-transitions as unobservable: this is consistent with switching from genetic units to material units (see above). A problem arises because tjk,x(h) may be significantly different from tkj,x(h), as transition probability, unlike cross-variograms, considers asymmetry, i.e.

transition probability takes into account direction-specific patterns (Carle & Fogg 1996). Therefore, it could be possible to derive a reference value of range for the cross-variograms from transition rates computed from bidirectionally-averaged transition probabilities, obtained by sampling FAKTS’ space in both directions for each axis (e.g. upwards and downwards along the vertical axis); this would be of range. Assuming the coincidence in transition counts in both directions, given by

x

Assuming this, the relationship between indicator cross-variogram and transition probability given by:

Thus, the tangent of the cross-variogram at lag zero is equal to:  p_jr_jkh

This means that the lag value at the intersection between the sill of the indicator

This approach only generates approximate models of cross-variograms, but it would only be used whenever lack of direct data precludes the calculation of cross-variograms (e.g. actual lack of data or need to generate a training image). An example application whereby FAKTS is used to derive indicator cross-variogram

parameters for material units defined on facies unit types, referring to horizontal directions, is presented in tables 6.5 and 6.6.

Figure 6.10: (A) example spatial transition probability between categories j and k as a function of lag h (tjk(h)) in a given direction, and corresponding cross-variogram (γjk(h)): the tangents to the curves at lag zero, the sills and their intersections are represented; the lag value at the sill-tangent intersection constrains the minimum value of cross variogram range; (B) analytical cross-variogram obtained from the sill and estimated range values derived from category proportions and transition rates;

(C) the sampling of the analytical variogram at given lag spacing yields cross-variogram values that can be used as input in plurigaussian simulations.

Table 6.5: FAKTS-derived indicator cross-variogram parameters for material units

corresponding to five selected facies unit types, referring to the cross-valley direction;

range corrected in excess of 10% of calculated tangent/sill intersection lag value.

INDICATOR CROSS-VARIOGRAM parameters – direction X (cross-valley/ strike)

St Sp Sr Sh

Range Sill Range Sill Range Sill Range Sill St Tab. 6.4

Sp 33 -0.0050 Tab. 6.4

Sr ^{18 -0.0155 15 -0.0024 Tab. 6.4}

Sh 17 -0.0705 1.5 -0.011 3.5 -0.0347 Tab. 6.4 Fl 138 -0.0033 4.1 -0.0005 22 -0.0016 66 -0.0073

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Table 6.6: FAKTS-derived indicator cross-variogram parameters for material units

corresponding to five selected facies unit types, referring to the along-valley direction;

range corrected in excess of 10% of calculated tangent/sill intersection lag value.

INDICATOR CROSS-VARIOGRAM parameters –

To fully constrain a plurigaussian simulation, it is necessary to choose a model that describes the spatial relationships between the categories. Given that FAKTS describes the spatial relationships between genetic units in terms of transitions, transition frequency matrices can be derived from the database for every type of genetic unit (and can be related to material units by setting diagonal values as zero) in order to obtain a quantitative description of spatial relationships (e.g. transition patterns from Markov analysis) with which to inform the choice of an appropriate model of contact relations.

Other pixel-based methods that consider the spatial cross-correlation between the categories that need to be modelled describe the spatial structure by using Markov chains, quantifying spatial relationships using transition probabilities (Carle 1997a;

1999; Elfeki & Dekking 2001). A popular transition-probability/Markov-chain geostatistical simulation method is implemented in the software T-PROGS (Carle 1999). The T-PROGS package allows the calculation of three-dimensional Markov-chain models of spatial variability that can be used in sequential indicator simulations (SIS) that are iteratively adjusted – in terms of matching simulated and modelled transition probabilities – by applying a simulated quenching (zero-temperature annealing) algorithm (Carle 1996; 1997b) to generate a geostatistical realization of categorical variables. In comparison to variogram-based geostatistical methods, this transition-probability approach simplifies the link between observable attributes and model parameters. T-PROGS simulations are easily constrained by

proportions, mean dimensions and juxtapositional tendencies; the creation of variograms through either curve-fitting or modelling on FAKTS-derived data (as above) is not needed. In this method, spatial variability is incorporated in form of transition probability instead of indicator cross-variogram, thereby permitting the representation of asymmetrical correlation structures (e.g., fining-upward trends) (Carle & Fogg 1996). T-PROGS allows the generation of each 1D Markov chain following five different possible approaches, each of them leading to a transition rate matrix that defines the matrix exponential form of the Markov chain model (Carle 1999). After querying FAKTS in order to obtain mean dimensional parameters and transition counts (and then converting output into embedded transition frequencies or probabilities) for a given direction, the entries of the transition rate matrix for that direction can be derived by prescribing off-diagonal (cross-) transition rates by dividing embedded transition probability/frequency values by mean dimension and diagonal (auto-) transition rates by mean dimension (since:

categories, which can again be derived from FAKTS. Due to transition rate matrix properties, there is no need to specify row and column entries for one of the categories in the rate matrix: it is convenient to consider this “background” category as the material that “fills in the space” not occupied by other categories (e.g. low-energy fine-grained floodplain sediments).

Given (i) that direct data from which spatial transition probabilities can be computed are often lacking in the horizontal (strike and dip) directions, and (ii) that the advocation of Walther’s law (cf. Carle & Fogg 1997) is not advisable in fluvial depositional contexts (e.g. discrepancies in vertical and lateral transition probabilities testify to a tendency of vertical stacking of abandoned channel-fills on aggradational channel-fills that has no counterpart in the lateral directions), FAKTS can provide important quantitative information with which to condition the Markov models of lateral spatial variability. Examples of FAKTS-derived transition probability matrices for material units defined on facies unit types, referring to vertical and horizontal directions are presented in tables 6.7, 6.8 and 6.9.

In contrast to the SIS approach, plurigaussian simulations and T-PROGS take into account spatial relationships: this means that it is possible to reliably simulate more than two indicators, making it possible to work more confidently with material units defined on the basis of FAKTS’ architectural elements and facies units.

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Table 6.7: FAKTS-derived transition-probability matrix for material units (no embedded self-transitions) based on 15 selected facies unit types, referring to the vertical (upwards) direction; lower units in rows and upper units in columns; values based on 6562 embedded transitions.

Z Fl Fm Gcm Gh Gmm Gt P Sd Sh Sl Sm Sp Sr Ss St

Fl - 0.02 0.03 0.00 0.00 0.00 0.03 0.01 0.19 0.06 0.15 0.13 0.17 0.07 0.14 Fm 0.00 - 0.03 0.00 0.00 0.00 0.04 0.03 0.23 0.02 0.20 0.07 0.24 0.01 0.13 Gcm 0.02 0.02 - 0.01 0.00 0.05 0.01 0.03 0.18 0.07 0.18 0.17 0.02 0.06 0.22 Gh 0.04 0.00 0.00 - 0.00 0.00 0.00 0.00 0.20 0.20 0.08 0.00 0.00 0.16 0.32 Gmm 0.00 0.00 0.00 0.00 - 0.00 0.00 0.00 0.18 0.18 0.09 0.09 0.00 0.18 0.27 Gt 0.00 0.00 0.12 0.00 0.04 - 0.00 0.00 0.15 0.12 0.23 0.08 0.00 0.08 0.19 P 0.04 0.14 0.01 0.00 0.00 0.00 - 0.13 0.14 0.00 0.25 0.06 0.07 0.00 0.17 Sd 0.02 0.05 0.00 0.00 0.00 0.00 0.10 - 0.17 0.15 0.25 0.13 0.05 0.02 0.08 Sh 0.05 0.08 0.02 0.01 0.00 0.00 0.02 0.04 - 0.10 0.18 0.12 0.18 0.05 0.14 Sl 0.03 0.01 0.02 0.00 0.00 0.00 0.00 0.07 0.19 - 0.24 0.14 0.04 0.05 0.19 Sm 0.06 0.11 0.03 0.00 0.00 0.00 0.03 0.05 0.23 0.15 - 0.13 0.05 0.03 0.11 Sp 0.05 0.04 0.04 0.00 0.00 0.00 0.01 0.05 0.20 0.12 0.15 - 0.08 0.03 0.22 Sr 0.14 0.21 0.01 0.00 0.00 0.00 0.02 0.02 0.27 0.03 0.09 0.07 - 0.01 0.14 Ss 0.03 0.00 0.02 0.00 0.00 0.01 0.00 0.04 0.27 0.22 0.06 0.08 0.03 - 0.23 St 0.02 0.05 0.05 0.01 0.00 0.00 0.01 0.04 0.22 0.16 0.13 0.15 0.11 0.05 -