Links between External drift, Bayesian kriging, Collocated Cokriging
by T.Coléou (CGG-PTS Norge)
This report contains kriging definitions, equations and results for all kriging techniques which are implemented in the FastGeoTie software. These are most of the known kriging methods used for mixing hard data (e.g.: wells) and soft data (e.g.: relationship derived from seismic). Comparisons between methods are made. Among them, the link between External Drift and Collocated Cokriging which, to my knowledge, cannot be found in the literature.
As all equations are described, it can be used as a reference for the methodology.
This report contains also caracteristics of the different kriging methods, it can be used as a practical guide to kriging. The various kriging comparisons can be used to design software test procedures for future version of the software.
Notations
- Z is the variable under consideration.
- Z1 is the main variable under consideration when several are in use.
- Z2 is the secondary variable under consideration when several are in use.
- N is the number of data points where Z is known.
- a is one of the data points.
- x is the point where Z is to be estimated.
- Zx* is the estimated value of variable Z at point x.
- Za is the value of variable Z at point a.
- Z1a is the value of variable Z1 at point a.
- la is the kriging weight for the point a.
- Cab is the covariance of variable Z between point a and point b
- C12ab is the cross-covariance of variables Z1 and Z2 between point a and point b
- Xa is the value of co-ordinate X at point a.
- Ya is the value of co-ordinate Y at point a.
Simple Kriging
Estimate: linear combination of the N data values.
Constraints: none.
Kriging system (N equations):
Comments: This is the basic kriging system. For stationary models of variograms, the estimated value is 0 when distances to data points are beyond the range.
Known-Mean Kriging
Estimate: linear combination of the N data values. where m is known
Constraints: none.
Kriging system (N equations):
Comments: Known-Mean Kriging is a Simple Kriging on a variable which has been centered on a value. This is the usual implementation of Simple Kriging. For stationary models of variograms, the estimated value is the given mean m when distances to data points are beyond the range. It is useful when kriging residuals and forcing the correction to be 0 away from the data points.
Ordinary Kriging
Estimate: linear combination of the N data values.
Constraint: no bias condition, also called universality condition, can be considered as a polynomial drift of degree 0.
Kriging system (N + 1 equations):with solution
Comments: Ordinary Kriging is the most frequent implementation of kriging. A by-product of ordinary kriging is the "geo-mean", function of the variogram, which estimates the average of the data points with filtering of their spatial redundancy. It is a weighted average, giving to each data point a weight proportional to its influence in the knowledge of the field. It is the unbiased version of the statistical average. It is also the trend or drift (a constant value) to which the estimate is attracted away from the data points.
In the stationary case, the estimated value is the "geo-mean" when distances from data points are larger than the range of the variogram model.
With a pure nugget effect as a variogram model, the estimated value away from data points will be the statistical average of the data points, which is equal to the "geo-mean" in that case.
The "geo-mean" will also be equal to the statistical average of the data points when the distance between all data points is larger than the range of the variogram model.
The "geo-mean" is a declustered version of the statistical average of the data points. It is a better estimate of the average value of a variable (interval velocity, porosity ...)
Kriging with Polynomial Drift (Intrinsic Random Functions of order k)
Estimate: linear combination of the N data values.
Constraints: one per term of polynomial drift.
- Drift of degree 0: 1 term drift = constant.
- Drift of degree 1: 3 terms drift = a + b X + c Y.
- Drift of degree 2: 6 terms drift = a + b X + c Y +d X Y + e X2 + f Y2.
Example for a polynomial drift of order 1 (plane in the X-Y domain):
Kriging system for a polynomial drift of order 1 (N + 3 equations):
Comments: Also known as Intrinsic Random Function of Order k, kriging with polynomial drift, for order larger than 0, accepts different models of covariance, called generalised covariances.
A by-product of kriging with polynomial drift is the calculated drift or "geo-regression", function of the variogram, which estimates the average plane fitted through the data points ( drift = a + b X + c Y ) with filtering of their spatial redundancy. It is a weighted regression, giving to each data point a weight proportional to its influence of the knowledge of the field. It is the unbiased version of the statistical linear regression on the co-ordinates.
With a stationary model of variogram, for points at distances from data points larger than the range, the estimated value is the "geo-regression" value.
With a pure nugget effect as a variogram model, or with a stationary model where the range is smaller than the smallest distance between data points, the estimated "geo-regression" will be the statistical linear regression of the variable Z against X and Y co-ordinates.
Kriging with External Drift
Estimate: linear combination of the N data values.
Constraints: one per external function used as drift, here only one is used.
Kriging system (N + 2 equations):
Comments: Kriging with external drift is not limited to one external variable.
A by-product of kriging with external drift is the calculated drift or "geo-regression", function of the variogram, which estimates the average regression fitted through the data points ( drift = a + b S ) with filtering of their spatial redundancy. It is a weighted regression, giving to each data point a weight proportional to its influence of the knowledge of the field. It is the unbiased version of the statistical regression.
With a stationary model of variogram, for points at distances from data points larger than the range, the estimated value is the "geo-regression" value.
With a pure nugget effect as a variogram model or with a stationary model where the range is smaller than the smallest distance between data points, the estimated "geo-regression" will be equal to the statistical linear regression of the variable Z against the variable S.
Bayesian Kriging
Estimate: linear combination of the N data values, after removal of a computed trend or drift : a* + b* S under constraint on a and b.
Constraints: a-priori knowledge of a and b, their uncertainties and their cross-correlation.
- a = E[ A ] a-priori intercept of the "geo-regression"
- b = E[ B ] a-priori slope of the "geo-regression"
- VarA = Var[ A ] variance of a the lower the variance, the higher the constraint
- VarB = Var[ B ] variance of b the lower the variance, the higher the constraint
- CAB = Cov[ A,B ] cross covariance between a and b
Kriging system: in two steps, determination of a* and b*, then Simple Kriging of the residuals.
Then it is a Simple Kriging system for the estimation of the residuals:
Comments: Bayesian Kriging is in theory not limited to one external variable but in practice it is limited to one as it is quite impossible to determine reliable a-priori information on several variables along with all their cross-correlations.
Within the Bayesian Kriging approach, the drift or "geo-regression" is calculated explicitly. It is a weighted regression, giving to each data point a weight proportional to its influence of the knowledge of the field. It is function of the chosen variogram model and of the constraints provided on each of the parameters of the linear "geo-regression".
One should note that the a-priori information, when there is some redundancy within the data set (e.g.: a lot of wells in a mature field), should not be directly deduced from standard statistical regressions, which are biased due to the preferred sampling.
When there is limited local information but when regional information is available (e.g.: in a satellite field), this technique is interesting. It enables to introduce the external regional knowledge into the ill-constrained local problem.
When constraints are ineffective, Bayesian Kriging leads to similar results than External Drift: it can be considered as a generalisation of External Drift method.
Collocated Cokriging
Estimate: linear combination of the N data values for the main variable Z1 and of the N + 1 data values for the secondary variable Z2.
Constraints:
Kriging system (N + N + 3 equations):
Comments:With an intrinsic model of corregionalisation (also called Markov model), all variograms and cross-variogram are proportional for all distances. The model is completely described with the parameters of C and with rho:
This Markov model is the one recommended in the litterature as it is the only one which can easily be infered when mixing very heterogeneously sampled data (tying seismic data to well data). In that case the recommended procedure is to fit a variogram model on the seismic data. Once this is made, to calculate the coefficient of proportionality to be applied by calculating the coefficient of correlation between the variables. This is a dangerous procedure as it implies the use of a regression instead of a "geo-regression". The regression can be biased due to preferential sampling.
It is possible to add an extra nugget effect component, on the second variable only. This will act as a filter on the secondary variable during the estimation.
Equivalence between Collocated Cokriging and External Drift
The External Drift estimate can be written as a sum of residuals and drift:
Z = Y + a S + b
where Y represents the residuals and S the secondary variable.
If we write the External Drift estimate as:
Collocated Cokriging is written:
If we try to identify the different terms, we have:
which gives:
With an intrinsic model of corregionalisation, or Markov model, the Collocated Cokriging can be written as:
When introducing the equation [1] we obtain:
This is a kriging system, with a unique solution, if the second and third equations disappear, leading to:
When choosing the coefficient of proportionality rho between the two models to the coefficient a of the slope of the "geo-regression" given by External Drift method, normalised by the standard deviations, Collocated Cokriging is strictly equivalent to External Drift, with a very high computing overhead.
When choosing the coefficient of proportionality rho between the two models, the slope a of the regression applied is fixed, and Collocated Cokriging is equivalent to the Ordinary Kriging of the residuals. Identical results can be obtained at a fraction of the time when using Bayesian Kriging with fixed slope a and no constraint on the intercept.
Relationships between the different Kriging Methods
The estimate is always a linear combination of the data values, with or without constraints on the kriging weights. Away from the influence of the data points, the estimated value is attracted to the given trend or drift.
Kriging Method trend or drift
- Simple Kriging constant zero value
- Known-Mean Kriging given constant mean value
- Ordinary Kriging calculated constant geo-mean
- Kriging with polynomial drift calculated geo-regression on co-ordinates
- Kriging with external drift calculated geo-regression on external variable
- Bayesian Kriging calculated geo-regression on external variable under constraints
- Collocated Cokriging slope of regression on variable given by the coefficient of proportionality of the cross-correlation model
When a trend is calculated, the different kriging methods are equivalent to a Simple Kriging of the residuals once the trend has been subtracted from the variable. In all cases, except with Bayesian Kriging, the Simple Kriging of the residuals is equivalent to an Ordinary Kriging of the residuals as these residuals have a "geo-mean" equal to 0.
There is a relationship between Simple Kriging and Ordinary Kriging. The Simple Kriging weights are the Ordinary Kriging weights less their contribution to the kriging of the mean.
Ordinary Kriging can be seen as a Kriging with polynomial drift where the polynomial has a degree 0 (a constant).
When using co-ordinates X and Y as external drifts, Kriging with external drift is equivalent to kriging with a polynomial drift of order 1.
Bayesian Kriging is a generalisation of External Drift. Without constraints, results are identical.
Collocated Cokriging with Markov model is equivalent to an Ordinary Kriging of the residuals when the drift given by the cross-correlation factor has been subtracted.
Bayesian Kriging is equivalent to Collocated Cokriging when the chosen slope is fixed and identical to the one for Collocated Cokriging and when there is no constraint on the intercept.
Bayesian Kriging can replicate all other methods:
- slope fixed to 0, intercept fixed to 0 : Simple Kriging
- slope fixed to 0, intercept free : Ordinary Kriging
- slope free, intercept free
1