Abstract:
The present invention relates to a method for determining a truncation diagram for a pluri-Gaussin geological parameter estimation associated with a zone of a real subsoil. The method comprises receiving an auxiliary variable describing the zone, computing a normalized variable based on the auxiliary variable and determining a weighted probability map for the normalized variable, each point of the weighted probability map being in correspondence with a point, of the truncation diagram. Moreover for each point of the truncation diagram, the method further comprises determining an associated geological parameter based on the probability value of each point of the weighted probability map. The method is also used to adapt truncation diagram to global or local target, proportions of the facies model to built.

Description:
RELATED APPLICATIONS 
       [0001]    The present application is a National Phase entry of PCT Application No. PCT/EP2013/057743, filed Apr. 12, 2013, which claims priority from EP Patent Application No. 12305751.5, filed Jun. 26, 2012, said applications being hereby incorporated by reference herein in their entirety. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    The present invention relates to geological parameters determination and especially to Gaussian and pluri-Gaussian simulations of lithologies. 
         [0003]    The approaches described in this section could be pursued, but are not necessarily approaches that have been previously conceived or pursued. Therefore, unless otherwise indicated herein, the approaches described in this section are not prior art to the claims in this application and are not admitted to be prior art by inclusion in this section. Furthermore, all embodiments are not necessarily intended to solve all or even any of the problems brought forward in this section. 
         [0004]    The estimation of an oil reservoir or a mining deposit usually comprises determination of geological parameters such as the facies composition of a subsoil. 
         [0005]    Truncated Gaussian simulations have been first introduced in 1987 and have been commonly used in geological simulation to determine realistic facies repartition. These simulations are based on a random field truncation, therefore a truncation order that should be defined for each simulation. 
         [0006]    With truncated Gaussian simulations, lithofacies are sorted for truncation (i.e. stratigraphic sequence): lithofacies that are not successors in the stratigraphic sequence will never be in contact. This feature has been proved to be too restrictive: for example, the facies may be impossible to order as there is no such order in the subsoil; moreover specific facies may be in contact with more than two facies. 
         [0007]    To overcome these limitations, pluri-Gaussian simulations were proposed in 1994 and 1996. The basic idea behind pluri-Gaussian simulations is to start out by simulating more than one Gaussian random fields at every grid point in the study domain and to use some simple rule to convert these Gaussian values into lithotype indicators (or more generally into geological categorical property or facies). 
         [0008]    In pluri-Gaussian models, a set of p Gaussian random fields are used to define the lithofacies structure. Most of the time p is set to 2 but similar works may be made with different values of p. The multi-variate Gaussian field is transformed into a multi-variate uniform field. Lithofacies are then assigned according to a truncation diagram (named also rock type rule or facies assignation diagram). Typical truncation diagrams separate the bi-variate uniform domain along vertical and horizontal lines. The precise layout or these lines defines the type of the chosen model. 
         [0009]    These simulations are intended to model complex geology with different structure orientations and heterogeneous deposits (channels, reefs, bars, differently oriented facies, sets of conjugate veins or ore types where geological constraints apply, etc.). They may provide realistic and detailed images of internal structure. 
         [0010]    These simulations may allow controlling the facies relationships and boundaries when dealing with complex geometrical configurations. 
         [0011]    Nevertheless, such Gaussian or pluri-Gaussian simulations have drawbacks. 
         [0012]    For example, it may be difficult to determine the relationship between Gaussian variables and physical processes. Moreover, the determination of the truncation diagram may not be easy and intuitive. 
         [0013]    If non-stationary parameters are present in the model, the update of the truncation diagram, according to the non-stationary parameters, may be very complex. Therefore, person skilled in the art would prefer creating simplistic truncation diagrams (for instance, with rectangle domains) in order to be able to adapt them to target proportions. 
         [0014]    There is thus a need for a method to determine complex (but adequate) truncation diagram in order to efficiently model and estimate facies repartition in a subsoil. 
       SUMMARY OF THE INVENTION 
       [0015]    The invention relates to a method for determining a truncation diagram for a pluri-Gaussian estimation of geological parameters associated with a zone of a real subsoil, the truncation diagram having a plurality of points. The method comprises:
       for each geological parameter in a plurality of geological parameters:
           receiving an auxiliary variable describing the zone;   computing a normalized variable based on the auxiliary variable;   determining a weighted probability map for the normalized variable, the probability map having a plurality of points, each point of the weighted probability map being in correspondence with a point of the truncation diagram, each point of the weighted probability map having a probability value;   
           for each point of the truncation diagram:
           determining an associated geological parameter based on the probability value of each point of the weighted probability map, each point of the weighted probability map being in correspondence with said point of the truncation diagram.   
               
 
         [0022]    An accurate determination of a truncation diagram is important in order to model subsoil in an adequate manner. If geologists or well engineers try to determine a truncation diagram, in most case, they simplify the diagram (to be able to handle its complexity) and achieve a non-accurate determination. 
         [0023]    In addition, this method enables the determination of a plurality of zones in a same area in order to adapt the truncation diagram based on the target proportions at the exact location in a zone in the plurality of zones: therefore, it is possible to determine a truncation diagram function which returns a truncation diagram based on coordinates in a zone and/or the target proportions at said coordinates. 
         [0024]    Therefore, for industrial benefits, there is a true need to determine efficiently truncation diagram. 
         [0025]    Auxiliary variables may be for instance, seismic attributes such as amplitude attributes (mean amplitude, average Energy, root mean square amplitude, maximum magnitude, AVO attributes, anelastic attenuation factor), time/horizon attributes (coherence, dip, azimuth, curvature), or log measurement attributes at wells (bulk density log, neutron log, photo-electric factor log, . . . ). 
         [0026]    Auxiliary variables may also be control points defined by the user. 
         [0027]    The auxiliary variable may be an n-variate variable, n&gt;1. 
         [0028]    The method may be used to adapt truncation diagram to global or local target proportions of the facies model to built. 
         [0029]    In a possible embodiment, for each geological parameter in a plurality of geological parameters, the auxiliary variable may comprise two one dimensional components. 
         [0030]    Hence, the auxiliary variable U may have two components V and W (i.e. U=(V,W)) with V and W being real values. 
         [0031]    The variable U may be represented in a two-dimensional domain. This variable U may returns real values (e.g. U(10, 0.5)=5.7). 
         [0032]    In addition, the normalized variable may be computed according a uniform transform. 
         [0033]    The transformation may also be a linear transform, a Gaussian transform or any other transforms, which returns a [0,1]×[0,1] domain. 
         [0034]    Determining a probability map may comprise computing kernel regression estimate of the normalized variable. 
         [0035]    From the simultaneous knowledge of the auxiliary variable and the lithofacies at data points (x 1 , . . . x n ) (e.g. measured at wells), a kernel regression estimate of these conditional probabilities may be 
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         [0000]    where i−1 . . . n is an index on all data and I [A]  is the indicator function of event A, equal to 1 if A is verified and equal to 0 otherwise. The function K h (x) may be the kernel function 
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         [0000]    , or the BiSquare kernel, or the Epanechnikov kernel. 
         [0037]    According to another embodiment, determining a weighted probability map may comprise weighting each probability value of a probability map with a determined coefficient. Such weighting may ensure that all probability values of the probability map is in the domain [0,1]. The weighting may be equal to 1 if the probability values are already in the domain [0,1]. 
         [0038]    The weight used for determining a weighted probability map may be function of coordinates of the zone. 
         [0039]    Therefore, it may be possible to adapt the truncation diagram according to local characteristics of the zone. 
         [0040]    Furthermore, for each point of the truncation diagram, determining an associated geological parameter may comprise:
       identifying a probability map in the plurality of weighted probability map for which the probability value of the point in correspondence with said point of the truncation diagram is maximum;   associating the geological parameter of the identified probability map with said point of the truncation diagram.       
 
         [0043]    The truncation diagram may also be referred to as “assignation diagram” or a “rock-type rule”. 
         [0044]    In order to verify specified proportion of geological parameter (e.g. lithofacies) it is possible to adapt the weights of each probability maps, for instance, by iterations or by a Marquardt regularization. 
         [0045]    Geological parameters may be facies or lithotypes. 
         [0046]    Another aspect of the invention relates to a method to determine estimated geological parameters in a model of a real subsoil. The method may comprise, for at least one zone of the model:
       determining a weight value for each geological parameter in the plurality of geological parameters;   determining truncation diagram according to the determined weight values and according to the method described above;   computing estimated geological parameters of the at least one zone using a pluri-Gaussian truncated estimation.       
 
         [0050]    The determination of the weight value may be done using for instance a iterative process like Markard algorithm. 
         [0051]    Yet another aspect of the invention relates to a device for determining a truncation diagram for a pluri-Gaussian geological parameter estimation associated with a zone of a real subsoil, the truncation diagram having a plurality of points. The device comprises:
       an input interface for receiving an auxiliary variable for each geological parameter in a plurality of geological parameters describing the zone;   a circuit to compute a normalized variable based on the auxiliary variables;   a circuit to determine a weighted probability map for each normalized variable, the probability map having a plurality of points, each point of the weighted probability map being in correspondence with a point of the truncation diagram, each point of the weighted probability map having a probability value;   a circuit to determine, for each point of the truncation diagram, an associated geological parameter based on the probability value of each point of the weighted probability map, each point of the weighted probability map being in correspondence with said point of the truncation diagram.       
 
         [0056]    A fourth aspect relates to a computer program product comprising a computer readable medium, having thereon a computer program comprising program instructions. The computer program is loadable into a data-processing unit and adapted to cause the data-processing unit to carry out the method described above when the computer program is run by the data-processing unit. 
         [0057]    Other features and advantages of the method and apparatus disclosed herein will become apparent from the following description of non-limiting embodiments, with reference to the appended drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0058]    The present invention is illustrated by way of example, and not by way of limitation, in the Figures of the accompanying drawings, in which like reference numerals refer to similar elements and in which: 
           [0059]      FIG. 1   a  is a representation of a first two-dimensional random variable; 
           [0060]      FIG. 1   b  is a representation of a second two-dimensional random variable; 
           [0061]      FIG. 1   c  is an example of a truncation diagram; 
           [0062]      FIG. 1   d  is a representation of a lithology map; 
           [0063]      FIG. 2   a  is a possible two-dimensional representation based on auxiliary variables for a geological parameter; 
           [0064]      FIG. 2   b  is a representation of a kernel density estimation for a one-dimensional variable; 
           [0065]      FIGS. 3   a  to  3   c  are examples of probability maps for three different facies; 
           [0066]      FIG. 4  is a representation of the computing of a truncation diagram based on two probability maps and depending on weights; 
           [0067]      FIGS. 5   a  to  5   b  are examples of truncation diagram computed based on the examples of density maps of  FIG. 3   a  to  3   c;    
           [0068]      FIG. 6  is a flow chart describing a possible embodiment of the present invention; 
           [0069]      FIG. 7  is a possible device that enables a possible embodiment of the present invention. 
       
    
    
     DESCRIPTION OF PREFERRED EMBODIMENTS 
       [0070]      FIG. 1   a  is a representation of a first two-dimensional random variable Z 1 . In a two-dimensional domain U−{[0, x max ], [0,y max ]}, it is possible to compute a random variable Z 1  having a coordinate (x, y) with x in [0, x max ] and y in [0,y max ]. 
         [0071]    This random variable Z 1  may be a continuous variable. 
         [0072]    Then, for each coordinate (x,y) in the domain U, the random variable Z 1  has a stochastic value. For instance:
       the stochastic value of point A is Z 1 (A)=0.35;   the stochastic value of point B is Z 1 (B)=0.5; and   the stochastic value of point C is Z 1 (C)=0.8.       
 
         [0076]      FIG. 1   b  is a representation of a second two-dimensional random variable Z 2 . In this example, random variable Z 2  is defined in the same domain U. 
         [0077]    The value of the variable Z 2  is different from the value of Z 1  for points having same coordinates:
       the stochastic value of point A is Z 2 (A)=0.45;   the stochastic value of point B is Z 2 (B)=0.2;   the stochastic value of point C is Z 2 (C)=0.05.       
 
         [0081]      FIG. 1   c  is an example of a simple truncation diagram with rectangular domains. This diagram  100  may be used to determine a lithology map. 
         [0082]    Four different facies are represented in this truncation diagram: facies  101 ,  102 ,  103  and  104  respectively represented by a rectangle or a rectangular domain. 
         [0083]    In order to determine a lithology map (as shown in  FIG. 1   d ), each point of the domain U={[0, x max ], [0,y max ]}, will be evaluated according to this truncation diagram. 
         [0084]    For instance, to determine facies associated with point A of  FIG. 1   d,  the following algorithm may be followed:
       determining the value Z 1 (A) of point A in  FIG. 1   a,  i.e. 0.35;   determining the value Z 2 (A) of point A in  FIG. 1   b,  i.e. 0.45;   identifying a point FA in the truncation diagram having coordinates (Z 1 (A), Z 2 (A)) i.e. FA(0.35, 0.45);   identifying the zone of the truncation diagram comprising point FA, i.e. rectangle associated with facies  104 ;   associating the facies of the identified zone with point A in  FIG. 1   d,  i.e. facies  104 .       
 
         [0090]    Same algorithm may be used for other points of domain U. 
         [0091]    For instance, for point B, the identified point in the truncation diagram is the point FB which is on an intersection point separating facies  101 ,  102  and  104 . Therefore, point B is also on intersection point separating facies  101 ,  102  and  104  in  FIG. 1   d.    
         [0092]    For point C, the identified point in the truncation diagram is the point FC which is on a separation line between facies  102  and  103 . Therefore point C is also on a separation line of facies  102  and  103  in  FIG. 1   d.    
         [0093]    This method described above may be referred to “Gaussian estimation for facies determination”. 
         [0094]      FIG. 2   a  is a possible two-dimensional representation based on two auxiliary variables for a geological parameter. 
         [0095]    Auxiliary variables may be for instance seismic attributes I p  and P r  (when available), or synthetic variables. 
         [0096]    For instance, a global variable may be defined a set of vector G=({right arrow over (x)} 1 , . . . {right arrow over (x)} n ). Facies information at a given point (for instance, thanks to wells and/or drill-holes) may allow creating a plurality of separated variables {G k } comprising values for each facies from the global variable G. For each k, G k  may comprise a subset of vectors in ({right arrow over (x)} 1  . . . {right arrow over (x)} n ), the subset of vector being representative of a given facies. 
         [0097]    The first component of vectors is, for instance, seismic attribute I p  and the second component is seismic attribute P r . 
         [0098]    Each point of the variable represented in  FIG. 2   a  ( 200 ,  201 ,  202  . . .  215 ) represents a value of the variable (or a vector). The repartition of points in the domain {[0,V max] , [0,W max ]} is not uniform. 
         [0099]    In order to quantify the density of the points, a kernel density estimate or kernel regression estimate may be computed. For instance,  FIG. 2   b  is a representation of a kernel density estimation for a one-dimensional variable. 
         [0100]    In  FIG. 2   b,  six points representative of a one-dimensional variable are disposed along the axis {right arrow over (x)}: x 1 , x 2 , x 3 , x 4 , x 5 , and x 6 . To determine a local density estimate, a kernel density estimator {circumflex over (f)} h (x) is centered on the abscissa of each point, wherein, for instance, 
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         [0000]    h&gt;0 and K(.) is the kernel (symmetric function such as uniform one, triangular one, bi-weight one, tri-weight one, Epanechnikov one, normal one, etc.). 
         [0101]    For point x 1  (respectively x 2 , x 3 , x 4 , x 5 , and x 6 ), a local density estimate  231  (respectively  232 ,  233 ,  234 ,  234 ,  235 ,  236 ) is placed so that the maximum of this local estimate corresponds to the abscissa of point x 1  (respectively x 2 , x 3 , x 4 , x 5 , and x 6 ). 
         [0102]    When all points (x 1 , x 2 , x 3 , x 4 , x 5 , and x 6 ), are associated with a local density estimate ( 231 ,  232 ,  233 ,  234 ,  234 ,  235 ,  236 ), the local estimates are summed to compute a global kernel density estimate  237 . 
         [0103]    Although  FIG. 2   b  is a representation of a density estimate for a one-dimensional variable, the generalization to a pluri-dimensional variable is possible thanks to the above disclosure. 
         [0104]      FIGS. 3   a  (respectively  3   b  and  3   c ) is schematic representation of a density estimate (or probability maps) in a bi-variate domain [U(I p ), U(P r )] representing a first facies (respectively a second and a third facies). 
         [0105]    This bi-variable U(I p , P r ) is an auxiliary variable (seismic parameters) describing a zone of a real subsoil. 
         [0106]    For instance, the density estimate (i.e. “probability value”) of the first facies FAC 1  (see  FIG. 3   a ) in point  301  is 0.15, in point  302  is 0, and in point  303  is 0.25. 
         [0107]    The density estimate of the second facies FAC 2  (see  FIG. 3   b ) in point  301  is 0.5, in point  302  is 0.05, and in point  303  is 0.25. 
         [0108]    Finally, the density estimate of the third facies FAC 3  (see  FIG. 3   c ) in point  301  is 0, in point  302  is 0.5, and in point  303  is 0. 
         [0109]    In addition, it is possible to transform the variable so that the values (i.e U(i,p) with p in [0, I max ] and I in [0, P max ]) of the represented variables are in the domain [0,1]. Gaussian transformation are suggested to transform the values in [0,1] domain but other transformations may also be used such as piecewise linear transformations, logarithmic transformations, etc. 
         [0110]    The above transformations (or only a selection of them) may be considered as, for an auxiliary variable, a computing of a normalized variable. 
         [0111]      FIG. 4  is a representation of the computing of a truncation diagram (element  403 ) based on two probability maps  401  and  402  (or “density map”) such as the probability maps of  FIG. 3   a,    3   b  or  3   c.    
         [0112]    The truncation diagram  403  is defined in a domain {[0,1], [0,1]} like the probability maps  401  and  402 . 
         [0113]    Probability map  401  represents facies FAC 1  and probability map  402  represents facies FAC 2 . 
         [0114]    As points  404  and  406  (respectively  405  and  406 ,  407  and  409 ,  408  and  409 ,  410  and  412 ,  411  and  412 ) have same coordinates in domain {[0,1], [0,1]}, it is possible to consider that point  404  (respectively  405 ,  407 ,  408 ,  410 ,  411 ) of a probability map  401  (respectively  402 ,  401 ,  402 ,  401 ,  402 ) is in correspondence with point  406  (respectively  406 ,  409 ,  409 ,  412 ,  412 ) of the truncation diagram  403 . 
         [0115]    Even if definition domains of truncation diagram or probability maps are different from {[ 0 , 1 ], [ 0 , 1 ]}, it is always possible to create a correspondence mapping between points of a probability map and between points of the truncation diagram according to specific correspondence rules. 
         [0116]    As described above, points of a probability maps have different probability values. For instance, point  404  (respectively  405 ,  407 ,  408 ,  410  and  411 ) is associated with the probability value V 404  (respectively V 405 , V 407 , V 408 , V 410 , and V 411 ). 
         [0117]    Probability values of points of probability maps may be weighted in order to modulate the importance of these probability maps. For instance, probability values of points of probability maps  401  (respectively  402 ) are weighted according to a weight w 1 =0.3 (respectively w 2 =0.7). 
         [0118]    Advantageously, the sum of weights of probability maps (i.e. w 1 +w 2 ) may be equal to 1. In another embodiment, all weights for all probability maps may be equals to 1. Once probability values of probability maps are weighted, the probability maps may be referenced as “the weighted probability maps”. 
         [0119]    To identify the associated facies with point  406  (respectively  409 ,  412 ) of the truncation diagram, weighted probability values (i.e. w 1. V 404  and W 2. V 405 ) of correspondent points (i.e.  404  and  405 ) in the probability maps  401  and  402  are compared. If the value of (V 404 . w 1 ) is greater than the value of (V 405 . w 2 ), then facies FAC 1  is associated with point  406 . Otherwise, FAC 2  is associated with point  406 . Similar determination may be computed with point  409  and  412 , or any other points of the truncation diagram. 
         [0120]      FIGS. 5a to 5b  are examples of truncation diagram computed based on the examples of probability maps of  FIG. 3   a  to  3   c.    
         [0121]      FIG. 5  a is a possible truncation diagram determined according the process described in  FIG. 4 :
       the probability values of probability map (representing facies FAC 1 ) of  FIG. 3   a  being weighted by the weight w 1 =0.4,   the probability values of probability map (representing facies FAC 2 ) of  FIG. 3   b  being weighted by the weight w 2 =0.3,   the probability values of probability map (representing facies FAC 3 ) of  FIG. 3   c  being weighted by the weight w 3 =0.3.       
 
         [0125]    Line  501  represents the separation line between facies FAC 1  and FAC 2  in the truncation diagram  500 . Line  502  represents the separation line between facies FAC 2  and FAC 3  in the truncation diagram  500 . Line  503  represents the separation line between facies FAC 1  and FAC 3  in the truncation diagram  500 . 
         [0126]      FIG. 5   b  is a possible truncation diagram determined according the process described in  FIG. 4 :
       the probability values of probability map (representing facies FAC 1 ) of  FIG. 3   a  being weighted by the weight w 1 =0.5,   the probability values of probability map (representing facies FAC 2 ) of  FIG. 3   b  being weighted by the weight w 2 =0.4,   the probability values of probability map (representing facies FAC 3 ) of  FIG. 3   c  being weighted by the weight w 3 =0.1.         
         [0130]    As the weight w 3  in  FIG. 5   b  is lower than the weight w 3  in  FIG. 5   a  (and as the sum w 1 +w 2 +w 3  being constant), the domain associated with facies FAC 3  in  FIG. 5   b  is smaller than the domain associated with facies FAC 3  in  FIG. 5   a.    
         [0131]    Therefore, it is possible to determine the proportion of the different facies by adapting the different weights. Iteration process may be used to determine the correct/adequate weights. 
         [0132]      FIG. 6  is a flow chart describing a possible embodiment of the present invention. 
         [0133]    This flow chart describes a possible method for determining a truncation diagram for a pluri-Gaussian facies estimation. 
         [0134]    Upon reception of raw information/data of a zone of a subsoil (message  600 ), this information/data is pre-computed (step  601 ) in order to separate data associated with the different facies of the subsoil. 
         [0135]    This pre-computation  601  enable the creation of a plurality of bi-variables L 1 (v,w), L 2 (v,w) . . . Ln(v,w) ( 602 ,  603  . . .  604 , etc.), each variable describing one facies of the zone of the subsoil. 
         [0136]    For each variable L 1 (v,w), L 2 (v,w) . . . Ln(v,w), the variable may be normalized (step  605 ) as described in relation of  FIGS. 3   a  to  3   c.    
         [0137]    Moreover, a probability map may be computed (step  607 ) according to a kernel regression estimate described for instance in relation of  FIG. 2   b  and  FIGS. 3   a  to  3   c.  An optional weighting (step  608 ) may be applied on probability values of the probability map as described in relation of the previous Figures. 
         [0138]    When all the weighted probability maps are computed (elements  611 ,  612  . . .  613 ), the truncation diagram (element  615 ) may be defined (step  614 ) according to the method described in relation of  FIG. 4 . 
         [0139]    Part of this flow chart can represent steps of an example of a computer program which may be executed by the device of  FIG. 7 . 
         [0140]      FIG. 7  is a possible embodiment for a device that enables the present invention. 
         [0141]    In this embodiment, the device  700  comprise a computer, this computer comprising a memory  705  to store program instructions loadable into a circuit and adapted to cause circuit  704  to carry out the steps of the present invention when the program instructions are run by the circuit  704 . 
         [0142]    The memory  705  may also store data and useful information for carrying the steps of the present invention as described above. 
         [0143]    The circuit  704  may be for instance:
       a processor or a processing unit adapted to interpret instructions in a computer language, or   an electronic card wherein the steps of the invention are described within silicon, or   a programmable electronic chip such as a FPGA chip (for          Field-Programmable Gate Array         ).       
 
         [0147]    This computer comprises an input interface  703  for the reception of data used for the above method according to the invention and an output interface  706  for providing the truncation diagram. 
         [0148]    To ease the interaction with the computer, a screen  701  and a keyboard  702  may be provided and connected to the computer circuit  704 . 
         [0149]    A person skilled in the art will readily appreciate that various parameters disclosed in the description may be modified and that various embodiments disclosed may be combined without departing from the scope of the invention. 
         [0150]    In particular, in the above description, reference is made to “facies”. This reference may also be, without changing the scope of the invention, “lithotype” or any geological characteristics. 
         [0151]    Moreover, auxiliary variables described in relation of Figure are bi-variables, but the variables may have a various number of components: the adaptation of above examples is rendered clear in view of the description. 
         [0152]    The above description has been disclosed with two-dimensional random variables but the adaptation with a different number of random variables (e.g. one dimensional or three dimensional random variables) is straight forward in view of the description. 
         [0153]    The embodiments above are intended to be illustrative and not limiting. Additional embodiments may be within the claims. Although the present invention has been described with reference to particular embodiments, workers skilled in the art will recognize that changes may be made in form and detail without departing from the spirit and scope of the invention. 
         [0154]    Various modifications to the invention may be apparent to one of skill in the art upon reading this disclosure. For example, persons of ordinary skill in the relevant art will recognize that the various features described for the different embodiments of the invention can be suitably combined, un-combined, and re-combined with other features, alone, or in different combinations, within the spirit of the invention. Likewise, the various features described above should all be regarded as example embodiments, rather than limitations to the scope or spirit of the invention. Therefore, the above is not contemplated to limit the scope of the present invention.