Sources of Non Stationarity in the Semivariogram

Transcription

1 Sorces of Non Stationarity in the Semivariogram Migel A. Cba and Oy Leangthong Traditional ncertainty characterization techniqes sch as Simple Kriging or Seqential Gassian Simlation rely on stationary assmptions of first and second order to describe in a nmerical manner the behavior of particlar variables of natral events sch as a mineral deposit. Bt natral events are non stationary phenomena. One of the most common soltions to this problem in practice is sb domaining that consist of separating the mineral deposit in sb grops that are more psedo stationary, generally based on the physical nderstanding of the natral event. Even when sb domaining has been carried on with all the available information it does not garantee that the inflence of the non stationary featres on them have been completely mitigated. In this docment a way to measre the effects of non stationarity over a domain is presented. It can be sed at each stage of the sb domaining process in order to verify how the non stationary conditions still affects the sb - domains or to be aware of it in order to find appropriate strategies to model them. This is achieved throgh the interpretation of the featres that the experimental semivariogram calclated via the method of moments captre from the dataset that represents a particlar domain. Introdction The SK system relies on the semivariogram model to describe the spatial strctre of the conditioning data and the semivariogram model is a fnction that describes the spatial continity of a RF that obeys the intrinsic hypothesis and is first and second order stationary. The conditioning data shold be part of a realization of that RF. Then the SK system will estimate the parameters of the conditional distribtions nder those conditions. Since SK minimizes the variance of the estimation error there is no better approach for it. Recall SGS relies on the SK system to get the realization maps. Unfortnately the datasets sampled from natral events are non stationary, the domains are finite and the intrinsic hypothesis is vagely satisfied. A conseqence of estimating or simlating in sch environment is the conditional distribtions or the realization maps will be nrealistic so that they will not have any se for an economic evalation of mineral deposit. It is necessary to minimize the impact of the lack of those conditions before proceed to apply traditional estimation or simlation techniqes sch as SK or SGS. The impact of lack of the intrinsic hypothesis is small compared to the lack of the stationary conditions of the dataset and the domain. Prior of estimating or simlation the dataset shold be conditioned to psedo stationary conditions of first and second order. One way to do this is to perform sb domaining which is to separate the domain is sb sector where the behavior of the variable to be modeled is fairly close to stationary, proceed to model it in more stable domains and join all the parts or sectors to get the final model. There also other techniqes that modify the non stationary space e.g. trend removing and transform it to a more stationary environment for modeling, and others that deal with the intrinsic assmption where they se non parametric covariance strctres. Sb Domaining Uncertainty characterization of mineral deposits is an important stage in the mining indstry. Geostatistical methodologies are sed to bild nmerical models that characterize ncertainty parameters globally and locally which are reqired by mine planners for economic evalation of the potential mine project. Traditional geostatistic methodologies both for estimation and simlation rely on the kriging system. The kriging system is intended to minimize the estimation variance nder the assmptions of first and second order stationarity and the intrinsic hypothesis of the RF. Bt nfortnately the natral events that are involved in the genesis of mineral deposits are not stationary and modeling them nder sch assmptions cold lead to wrong reslts. An option to deal with this problem is to sb divide the mineral deposit in sb domains where the non stationary featres impact less. At this point it is important to focs on first and second order stationary featres sally in Gassian nits. Ordinary kriging is widely sed in mining for estimation prposes and in this case sb domaining focs mostly on the second order stationary assmption trying to delineate 3-

2 geologic nit types with high, medim and low grades of metal content taking into accont the proportional effect. On the other hand if simlation is reqired for economic analysis first and second order stationary featres in Gassian nits are focsed dring the sb domaining process. One of the common natral processes that cold not be modeled nder stationary assmptions is the transition of rock types sch as mineralized and non mineralized. This transition cold be hard of soft. A soft transition is the gradal change in the mean and local variability of the variables being analyzed as the position changes from one rock type to another (see Figre 4); the gradal change is not necessarily linear for the mean neither for the local variability. It cold also happen that the transition is abrpt which cold be a conseqence of strctral featres sch as falts or post mineral geologic events. The latter one wold be a scenario with presence of independent domains (see Figre 3). Most of the information that is involved in sb domaining is the rock type logging data. From a geologic perspective rock type logging cold be detailed or generalized. If detailed from a geostatistical perspective it wold be possible to find domains fragmented in many rock types. On the other hand if generalized two or more geostatistical domains cold be groped in a single rock type. Even if in the generalized domain the mean grade seems to be consistent along the domain the local variability cold be different and nder this condition assming stationarity wold be a bad decision (see Figre ). Figre shows sitable simplistic approach for the RV behavior to model natral events and in reality they are way too mch complex or intractable. z () () z 0 0 (a) (b) (c) N(0,) N(0,) z 0 (3) (4) z (b) (a) (b) 0 (a) (c) N(0,) N(0,) Mean Confidence Limits Figre, schematic D Gassian RV environments: () first and second order stationary; () first order stationary bt locally variable variance with two first and second order stationary domains (a), (c) and one first order stationary with variable local variance (b); (3) non stationary with two independent first and second order stationary domains (a) and (b); (4) non stationary with two first and second order stationary domains (a) and (c) and one non stationary domain (b). In estimation or simlation the aim is to assess ncertainty, it is achieved throgh the calclation of the mean and variance at the locations of nknown vales conditioned to available data from the domain. Estimating or simlating in a non stationary environment does not garantee that the obtained reslts are correct for modeling natral events. Since SK and conseqently SGS rely on stationary and intrinsic hypothesis assmptions, even when the estimated or simlated mean vales cold be somehow correct de to strong presence of conditioning data the estimation variance is not correct, it cold be nderestimated or overestimated according to the nderlying behavior of the variables in the domain (see Figre ). On the other hand the intrinsic hypothesis impact cold be considered small compared with the impact de to the lack of psedo stationary conditions. Semivariogram in a Stationary and Non Stationary Environments The variogram is the measre of variability between two point vales z( ) and z( ) at two different locations. (Barry 004) Analytically the variogram represent the variance strctre between two RV at different locations. It can be expressed as (Cressie, 993) the variance of the difference of two RV spatially located at locations and (). In order to calclate it many realizations of the two random variables 3-

3 Z( ), Z( ) wold be reqired, nfortnately in practice only one is available which is the conditioning data. ( ) { ( ) ( ) } γ, = var Z Z d, (), To overcome that problem the intrinsic hypothesis is assmed. (Matheron 97) The RF Z obeys the intrinsic hypothesis if the variance of the spatial difference Z() Z(+h) exist and is independent of location bt are fnction of the separation vector h. The variance of the spatial difference is the variogram expression () and the expected vale is a fnction of the separation vector and assmed to be a linear drift (3). Removing the linear drift from Z() for all the locations in the form R()=Z() m() the expected vale of the spatial difference of the residals R() become zero and the variogram expression can be written as the expected vale of the sqared spatial difference (4). { ( ) ( + )} = ( ) { ( ) ( + )} = ( ) var Z Z h γ h () E Z Z h m h (3) Removing the drift component from the RV Z() is eqivalent to make the RF first order stationary. Recall the RF R shold obey the intrinsic hypothesis. Then the variogram is expressed as follows: ( ) = ( ) ( ) R Z m { } ( ) ( ) ( ) γ h = E R R + h (4) The difference between expression () and (4) is in the mean vale of the RV. (Gneiting and others, 00) regardless of the definition of variogram or semivariogram de to the 0.5 factor, they distingish between centered and non centered variograms. Expression () is called centered and (4) is non centered. In order to link the semivariogram expression with the covariance and variance it is necessary to make assmptions of second order stationarity in R(). Then the semivariogram can be written in terms of the difference between the variance and the covariance (5). ( ) = σ c ( ) ( ) = ( ) γ h h (5) γ h σ ρ h (6) ( ) In eqation (5) γ(h) is the semivariogram, σ is the variance and c(h) the spatial covariance and in expression (6) ρ(h) is the spatial correlation coefficient. The semivariogram, the spatial covariance and the spatial correlation coefficient are fnction of the separation vector h. Recall the spatial covariance at h=0 is the variance therefore at no spatial correlation the semivariogram takes the vale of the variance. The spatial correlation coefficient at h=0 is. De to the Intrinsic Hypothesis the kriging estimation variance is dependent of the spatial configration of the conditioning data rather than variability patterns as a fnction of location in the domain. In a mining project analysis the estimation variance is often sed as a parameter to rank ncertainty so this vale cold be misleading if not sed properly. Since first and second order stationary assmptions and intrinsic hypothesis are assmed these three featres have an impact on the modeling of the reqired variables and ncertainty characterization in the domain. The natral event has to be assmed that follows a behavior like Random Fnction nder the three previos assmptions. Unfortnately the spatial featres of the natral events do not follow any of those assmptions and they have to be accommodated in a sitable manner so that the non stationary featres do not affect too mch the domain. Prior to modeling it is important to make the big domain as mch stationary as possible, some techniqes to achieve this are sb domaining and trend modeling. And to deal with non stationary covariance or semivariogram fnctions many approaches have been developed, see for example Sampson and Gttorp 99, proposed to move from an initial geographical dimension to a dispersion dimension where the intrinsic hypothesis is more sitable. 3-3

4 In practice expression () makes more sense in order to model natral events. The spatial variability is a fnction of the location and between two locations. One simple example is the presence of anisotropic behaviors where the spatial correlation in the direction of major continity is different to the minor s. The se of anisotropic ratios is an attempt to convert from a space where the intrinsic assmption is not valid to another one where it is. Bt natral events are more complicated than this. In addition to that there is no condition in the natral events that cold explain that the anisotropic behavior is elliptical. It is jst a simplistic approach to average local spatial continity strctres. Impact of non stationarity in experimental semivariogram calclation From expression (4) the estimator of the semivariogram is (7) also known as method-of-moments (Matheron 96) which can be also interpreted as the average vale of the orthogonal distances of the data pairs to the 45 line in the h scatter plot. ( h) ( ) ˆ γ ( h) = r( i) r( i + ) n h n h i= Where n(h) is the nmber of available pairs for the separation vector h. r( i ) and r( i +h) are the i-th data pair of the initial dataset at the head and tail of the separation vector h respectively. The experimental semivariogram plot consists of many different experimental semivariogram vales calclated for different separation vector (h i, i=,..,n) in ascending order of distance for one single direction for directional semivariogram plots and in all direction for omnidirectional semivariogram plots. In presence of sparse data the separation vector h is took with tolerances. In order to calclate the experimental semivariogram the initial dataset r 0 with mean m 0 and standard deviation s 0 is split in two parts or sb datasets r and r +h with mean m, m +h and standard deviation σ, σ +h respectively which correspond at the two extremes of the separation vector h. The representative dataset r h for each separation vector h consist of the nion of the two sb datasets r h =r Ur +h and for small separation distances the initial dataset and the representative dataset tend to be the same r 0 r h bt as the distance of the separation vector increases they tend to be different and r 0 r h since the sb datasets have less information and the experimental semivariogram vale is less reliable. Notice the initial dataset r 0 is not inflenced by any declstering weights, it is assmed to be fairly representative of the domain. At this point the available dataset for calclating the experimental semivariogram is limited or finite and the nmber of available pairs of samples for each configration of the separation vector is an isse now. Even thogh stationarity is assmed in a finite domain the mean and variance are not constant for the entire domain they are a fnction of how representative the sb datasets at the head and tail of the separation vector h are. These are some featres of finite domains that case problems with the stationary semivariogram featres, like the nmber of available samples for each separation vector and presence of sparse sampling data. From Figre when Gassian space is assmed the calclation of the semivariogram estimator for each separation vector h to be according to expression (7) shold prodce a symmetric h scatter plot and the data pairs shold follow a bivariate Gassian distribtion. In presence of enogh data the two sb datasets are expected to have similar, means eqal to zero, variances and marginal distribtion shapes closely to Gassian shape, eqal to one. In general the only parameter that is expected to change is the spatial correlation coefficient ρ(h). In this context the notice that the covariance plot is similar to the correlograms. Unfortnately those reqired conditions are not present in real data bt their impact can be qantified nmerically throgh the semivariogram expression. (7) 3-4

5 s(+h) (+h): tail m(+h) m() s() Figre, ideal h scatter plot of semivariogram calclation for a dataset which follows a Gassian distribtion. In Gassian nits the data pairs shold lie in a Gassian bivariate joint distribtion and the spatial continity is measred by the correlation coefficient which is a fnction of the separation vector. m, m +h are the means and σ, σ +h are the standard deviations of the distribtions of the sb datasets at the extremes of the separation vector h. The inflences of the mean and variances of the sb datasets at the two extremes of the separation vector h can be calclated from expression (7). Expanding the initial expression then adding and sbtracting their respective average vales of the sb datasets r and r +h it is possible to express (7) in terms of covariance, means and variances of the two sb datasets as follows: ( h) ( h) n (): head ˆ γ ( h) = + h + + h n ( r( i) ) r( i) r( i ) r( i ) ( ) i= n( h) n( h) ( r( i) ) ( r( i )) r i r i i= n i= ( ) ( ) = + + h + h n( h) ( h ) = σ + σ + c + [ m m + ] h h h The semivariogram expression is written as a fnction of the differences of standard deviations and difference of means of the sb datasets at locations and +h and the covariance (8). ˆ γ ( h ) = [ ] [ ] σ σ m m σ σ c + h + + h + + h h (8) Or the correlation coefficient (9) for a separation vector h. ˆ γ ( h ) = [ ] [ ] ( ) σ σ m m σ σ ρ + h + + h + + h h (9) From eqation (8) the semivariogram expression consists of for components. (a) Half of the average of the sqared difference of their respective standard deviations pls (b) half of the average of the sqared difference of their respective means pls (c) the prodct of the two standard deviations of the sb datasets of the head and tail of the separation vector mins (d) the covariance of the two sb datasets separated by the h vector. Notice that when the means and variances of the distribtions of the two datasets are fairly eqal (σ σ +h σ and m m +h m) the experimental semivariogram expression (8) relies on the spatial covariance and in expression (9) on the spatial correlation coefficient which are the stationary forms of the semivariogram. 3-5

6 In a stationary environment negative correlation coefficients are interpreted as if the initial dataset is affected by a large trend pattern (m m +h ) and are thoght to be the reason why the experimental semivariogram vales take vales greater than the variance of the dataset. (Gringarten & Detsch 00) Trends in the data can be identified from experimental semivariogram, which keeps increasing above the theoretical sill. In simple terms, this means that as distances between data pairs increase the differences between data vales systematically increase. Bt in practice this is not necessarily correct since the correlation coefficient ρ(h) is a fnction of the two sb datasets separated by h, the impact occr when differences in mean and variances of the two datasets are different (9). In presence of sparse data the data pairs are bilt sing tolerances in the search of pairs, this makes that for one data point for a particlar lag distance it cold be paired with more than one sample. In that case the distribtion of the sb dataset at the head of the separation vector h is the reslt of the weighting each sample location by the nmber of repetitions in the pairing de to the search tolerances. This vale is not fairly correct, it depends on the tolerances. For small tolerances the semivariogram calclation is more representative. Big tolerances tend to mask spatial featres of the dataset e.g. hiding of the anisotropic featres of the domain. Case Stdy The case stdy is a D nconditional simlated dataset of 000 data points reglarly spaced with a spherical semivariogram model of 5 nits of range. Three different trend cases were added to the initial dataset in order to describe them via the experimental semivariogram expression. The first trend component is a linear trend in the form y=ax+b with a negative slope along all the dataset. The second trend component is a symmetrical convex shape trend. For the third case the mean component remains the same bt the variance of the two halves of the dataset were modified (see Figre 3). Figre 3, Initial dataset inflenced by linear trend, parabolic trend and local variability in variances. Experimental semivariogram vales can be calclated sing expressions (8) or (9) so that the semivariogram vale is divided in three parts: ) the mean component, ) the variance component and 3) the stationary component. The semivariogram plots for each case are calclated for the half of the size of 3-6

7 the domain as a maximm separation distance in order to have enogh data pairs that represent the dataset. The contribtions of each component of the experimental semivariogram are represented as regions of different color (see Figre 5). Even when the initial dataset is spposed not to have any inflence of mean trend or variance trend it can be seen that there is presence of those components in the semivariogram plot. This effect can be seen comparing the two distribtions of the first and second half of the entire initial dataset (see Figre 4). Notice that even when the initial dataset lies on standard normal distribtion N(0,) it does not necessarily mean that locally is the same and that behavior can be captred by the experimental semivariogram. For comparison those variation in the initial dataset are assmed negligible, bt that will be captred by the experimental semivariogram. Figre 4, distribtion of the first 500 data points in the dataset (left) and 500 last data points (right) In three ot of the for cases the experimental semivariogram captre the first and second order non stationarity effect in their respective datasets. There is particlar case where the contribtion of the mean trend component cannot be seen in the experimental plots. It happens when the mean trend component is symmetric (the convex shape trend). The mean difference component occr when differences of the means appear as the separation distance increases and in this case since the trend component is symmetric the means of the sb datasets are cancelled (m =m +h ). On the other hand notice that nder that condition the only component that remains is the stationary component and the trend is captre nder the stationary conditions of semivariogram (experimental semivariogram vales greater than the variance of the initial dataset). This condition is interesting since in presence of directional symmetric mean trend shapes the stationary conditions of the semivariogram are valid again. This can also be seen in the h scatter plots for each case (see Figre 8, Figre 9, and Figre 0) Case Stdy The second case stdy was rn with real data for a vertical direction. The Amoco3d.dat dataset consist of 6 vertical wells with on average 53 samples per well (35 minimm and 66 maximm). There are a total of 3303 available data points in the dataset. Vertically the separation distance between the samples is one nit of distance. A vertical semivariogram plot of the porosity variable in normal score nits is calclated (Figre 6). Since the dataset is reglarly spaced in the vertical direction a lag distance interval of one nit of distance is chosen so there will not be overlap between pairs. Traditionally it cold be inferred that there is a presence of a mean trend since the vertical variogram does not reach the sill ntil the lag interval 38 where γ(38) = with 950 pairs ot of 34 bt there are two trend components that are captred for the dataset. The two trend components are present in the initial dataset for the vertical direction (variance trend and mean trend) (see Figre 7) and the variance for each location of the semivariogram start to be different from the lag 5. It is also important to verify how representative the sb datasets are for each lag interval. 3-7

8 Figre 5, experimental variograms for initial dataset and inflenced by linear trend, parabolic trend and locally variable variance. In the for plots the ble region represents the variation of the mean component, the red region the variation of the variance component and the green region the stationary component of the experimental semivariogram vale. Figre 6, distribtion (left) and vertical experimental variogram (right) for the NS vales of porosity Figre 7, experimental semivariogram of vertical direction for oilsand dataset 3-8

9 Conclsions The SK system is nbeatable calclating the parameters of the conditional distribtion at a location where the variable is nknown conditioned to previos existing data. Traditionally the SK system ses a spatial variability model that describes the behavior of a RF which is first and second order stationary and obeys the intrinsic hypothesis. The estimated parameters of the conditional distribtion then are also constrained to those conditions; additionally the conditioning dataset shold be part of the previosly mentioned RF. From those conditions the stationary assmptions are sally more important than the intrinsic hypothesis assmption since in presence of strong conditioning data the impact of the latter can be considered as negligible. In a geostatistical analysis the aim is to assess ncertainty. Usally parameters sch as mean and variance are enogh to describe it. In some cases when only the mean is calclated or considered important it cold lead to a misinterpretation of the geostatistical model reslts and conseqently lead to a wrong decision making process in a economic evalation of a potential mineral deposit project. In that case the geostatistical reslts are obtained bt not interpreted correctly. Since traditional estimation on simlation techniqes rely on stationary assmptions and sampled datasets from natral events are not necessarily stationary this approach is a sefl tool to measre nmerically the effects of non stationarity via experimental semivariogram calclation in the datasets and proceed to make decisions abot sb domaining or any other data processing techniqes in order to make the estimation or simlation more consistent with the natral event being modeled. Non stationary characteristics present in real datasets that represent natral events can be captred from it throgh interpretation of the many different featres of the experimental semivariogram tool comparing how different is the calclated from the ideal case. The semivariogram contains enogh information to describe the spatial continity beyond the stationary assmptions. References Chilès J.P. & Delfiner, P. Geostatistics: modeling spatial ncertainty. Wiley-Interscience, New York, 999. Genton M.G. Variogram Fitting for Generalized Least Sqares Using an Explicit Formla for the Covariance Strctre, Mathematical Geology, Vol. 30, No. 4, 998. Goovaerts, P. Geostatistics for Natral Resorces Evalation. Oxford University Press, New York, 997. Matheron G. The Theory of regionalized Variables and Its Applications, Ecole Nationale Spériere de Paris, Paris 97 Jornel A.G. & Hijbregts Ch.J. Mining Geostatistics. The Blackbrn Press, 978. Gringarten & Detsch, Variogram Interpretation and Modeling, Mathematical Geology, Vol. 33, No. 4, 00. Figre 8, h scatter plots for the initial case with no inflence of trend. 3-9

10 Figre 9, h scatter plots for the linear trend case Figre 0, h scatter plot for the parabolic shape trend case 3-0