• Estimation
• Facies Simulation Distribution
• Petrophysical Property Simulation
• Lecture 6 Quiz
Introduction
At any instant in time there is a single true distribution of a geological attribute. The true distribution is not available, but we do the best we can to map the true distribution given some sample data. From the need to map the true distribution as accurately as possible many interpolating algorithms were developed. The most common, anf the most usefull is kriging. Kriging is a locally accurate and smooth interpolator, appropriate for visualizing trends, but inappropriate for flow simulation where preservation of heterogeneity in the reservoir is important. An extension of the kriging algorithm is sequential simulation. Simulation is appropriate for simulation, and allows an assessment of uncertainty with alternative realizations. The first part of this Lecture is devoted to the kriging algorithm and kriging variance. The lecture then moves on to simulation of petrophysical properties and concludes with simulation of facies.
Estimation
Consider the problem of estimating the value of an attribute at any unsampled location u, denoted z*(u), using only sample data collected over the study area A, denoted by z(un) as illustrated in Figure 6.1.
Figure 6.1
The algorithm used to solve this problem was pioneered by Danie Krige and in recognition of his efforts the algorithm is called kriging. The kriging algorithms are a family of generalized least squares regression techniques that estimate z*(u) using sample data z(un). There are several different flavours of kriging, each addressing different needs. In general the the kriging equations are known as:
(6.1)
Where z*(u) is the estimator at location u, m(u) is the mean at location u, z(uα) is one of the α different data at location u used in the estimate, m(uα) is the mean at location u, and λα are the weights.The kriging equations state that the estimate is a weighted linear combination of the sample data, or more generally:
(6.2)
Reconsider figure 6.1 with equation 6.2 in mind. Equation 6.2 indicates that the estimator z*(u) is the weighted sum of the data, or mathematically:
(6.3)
There are a few goals that we strive for when choosing the weights:
1. closeness to the location being estimated, the estimator is equidistant from both knowns.
2. redundancy between the data values, the knowns lie on either side of the estimator. If the knowns were both on the same side as the estimator then it would be more difficult to make the estimate.
3. anisotropic continuity (preferential direction) 4. magnitude of continuity / variability
Figure 6.2. The kriging weights must consider redudancy of the data, the closeness of the data, and the direction and magnitude of continuity.
There is one other goal when estimating the unknown attribute: minimize the error variance. If the error variance is minimized then the estimate will be the best estimate. The error variance is the expected value of the difference between the known and the estimate and is defined by:
(6.4)
where z*(u) is the estimator, and z(u) is the true value. One obvious question raised by this equation is how can we determine the error if we do not know the true value? True, we do not know the true value, but we can choose weights that do minimze the error. To minimize the estimation variance take the partial derivative of the error variance (equation 6.4) and set to 0, but before taking the derivative equation 6.4 is expanded:
(6.5)
The result is an equation that refers to the covariance between the data points C(uα,uβ), and the data and estimatorC(u,uβ). A first this may seem like a problem because we have not dicussed the covariance between the data and the estimator, but we did discuss the variogram, and the variogram and the covariance are related. Recall that the variogram is defined by:
and note that the covariance is defined by (the covariance is not the squared difference whereas the variogram is):
so the variogram and the covariance are linked by:
(6.6)
where γ(h) is the variogram, C(0) is the variance of the data, and C(h) is the covariance.
This makes it possible to perform kriging in terms of the variogram instead of the covariance.
Continuing with the derivation of the kriging equations, we know that formula 6.5 must be minimized by taking the partial derivative with respect to the weights and set to zero:
...
setting to zero...
(6.7)
The result of the derivation in terms of the variogram is the same because both the variogram and the covariance measure spatial correlation, mathematically:
(6.8)
and the system of equatiuons in terms of the variogram is:
(6.9)
This is known as simple kriging. There are other types of kriging but they all use the same fundamental concepts derived here.
Discussion
There are a couple of motivations behind deriving kriging equaitons in terms of the covariance:
1. Its easier. Solving the kriging equations in terms of the variogram requires that the mean be carried throughout the derivation. It is easier to simplify in terms of covariance.
2. It is possible to have the variance at h=0 be zero with a variogram, this makes the matrix very unstable. The covariance is defined as the expected vaue of the difference, not the squared difference therefore the value of the covariance at h=0 is always large and hence the main diagonal in the matrix will always be large. Matrices that have small main diagonal elements such as when using the variogram are difficult for solution algorithms to solve due to truncation errors and so on.
3. it is eay to convert the variogram to covariance Implimenting Kriging
Once again, consider the problem of estimating the value of an attribute at any unsampled location u, denoted z*(u), using only sample data collected over the study area A, denoted by z(un) as illustrated in Figure 6.3. Figure 6.3 shows the estimator (the cube), and the data (z(un)). To perform kriging just fill in the matrices. For example, filling in the left hand matrix, entry 1,1, consider the variogram between points 1 and 1. The distance between a point and itself is 0, and thus the first entry would be the nugget effect. entry number 1,2, consider the distance h between points 1 and 2, read the appropriate variogram measure and enter it into the matrix. repeat for the all of the variogram entreis and solve for the weights λn.
Figure 6.3
Figure 6.4
Figure 6.5
The estimate is then calculated as:
(6.10)
The result is an estimate of the true value and the error associated with the estimate, as Figure 6.6 illustrates.
Figure 6.6
Kriging provides the best estimate but there are some issues:
The Pros and Cons of Kriging
Pros: Cons:
The “best” linear
unbiased estimator Smooths
Uses the variogram Does not honor the
Since kriging is a linear estimator it smooths, and thus reduces the heterogeneity in the model. This is acceptable for attributes that are already smooth but in most cases kriged outputs are not acceptable for mapping because the true heterogeneity is removed. Another issue that arises from smoothing is the failure of kriging to honor the histogram and the vaqriogram. This also a result of the smoothing effect of kriging. Kriging offers one solution to the estimation problem, but it offers the mean estimate for all points. There are other possibilities that are expressed in the error variance.
The Kriging Variance
Recall that the expanded kriging variance is:
(6.5) and that the kriging equation is:
(6.7) substituting equation 6.7 into equation 6.5:
(6.11)
Equation 6.11 is the kriging variance. Kriging may be considered spatial regression, that is, it creates an estimate that is smooth. There are many interpolation / estimation agorithms that construct smooth estimates. The advantage of kriging over other algorithms is that it provides quantification of how smooth the estimates are. The variance of the kriging estimate may be calculated as:
(6.12) substituting equation 6.7 into equation 6.12,
(6.13)
Equation 6.13 tells us the vaiance of the kriging estimator at location u, however, the variance is stationary, that is, the variance should be the same everywhere. Therefore there is a missing variance equal to:
(6.14)
Which is exactly the kriging variance. Thus the missing variance is the kriging variance. When kriging at a data location, the kriging variance is zero and there is no missing variance. When kriging with no local data hte kriging variance is C(0) and all of the variance is missing.
Petrophysical Property Simulation
Simulation addresses the cons of kriging and still makes use of the kriging algorithm and all of the good things about kriging. One of the important things that kriging does correctly and provides the motivation for sequential simulation is that it gets the covariance between the data and the estimator right. The issue that leads to sequential simulation is the fact that although kriging gets the covariance correct between the data and the estimate correct but it fails to get the covariance between the estimates. That is, rather than use the estimators as additional data the kriging algorithm simply moves on to the next estimator not including the covariance between the new estimator and the last.
Sequential simulation does just that. The first estimator is kriged with only data because that is all that there is, just data, no other estimates. The next estimate is kriged but the previous estimator is used as
data and its variogram is included in the algorithm.
This is sufficient motivation to proceed sequentially with estimation, but there is still the issue of the missing variance. Recall that the missing variance is the kriging variance:
(6.15)
This missing variance must be added back in without changing the variogram reproduction properties of kriging. This is done by adding an independent component with a zero mean and the correct variance to the kriged estimate:
(6.16)
(6.17)
The sequential simulation workflow is as follows:
1. Transform the original Z data to a standard normal distribution (all work will be done in
"normal" space). We will see later why this is necessary.
2. Go to a location u and perform kriging to obtain kriged estimate and the corresponding kriging variance:
• Draw a random residual R(u) that follows a normal distribution with mean of 0.0 and variance of σ2SK(u).
• Add the kriged estimate and residual to get simulated value:
Note that Z* (u) could be equivalently obtained by drawing from a normal distribution with mean Z*(u) and variance σ2SK(u).
• Add Y8 (u) to the set of data to ensure that the covariance with this value and all future predictions is correct. As stated above, this is the key idea of sequential simulation, that is, to consider previously simulated values as data so that we reproduce the covariance between all of the simulated values.
• Visit all locations in random order (to avoid artefacts of limited search).
• Back-transform all data values and simulated values when model is populated
• Create another equiprobable realization by repeating with different random number seed (Deutsch, 1999)
Why a Gaussian (Normal) Distribution?
The key mathematical properties that make sequential Gaussian simulation work are not limited to the Gaussian distribution. The covariance reproduction of kriging holds regardless of the data distribution, the correction of variance by adding a random residual works regardless of shape of the residual distribution (the mean must be zero and the variance equal to the variance that must be re-introduced), and the covariance reproduction property of kriging holds when data are added sequentially from any distribution. There is one very good reason why the Gaussian distribution is used: the use of any other distribution does not lead to a correct distribution of simulated values. The mean may be correct, the
the Gaussian distribution is used: the central limit theorem tells us that the sequential addition of random residuals to obtain simulated values leads to a Gaussian distribution. The construction of kriging estimates is additive. By construction, the residuals are independent and there are many of them. The only caveat of the central limit theorem that we could avoid is the use of the same shape of distribution, that is, we may avoid multivariate Gaussianity if the shape of the residual distribution was changed at different locations. The challenge would be to determine what the shape should be.
(Deutsch, 1999)
The Pros and Cons of Simulation
Pros: Cons:
Honors the histogram Multiple images
Honors the variogram Conceptually difficult to understand
Quantifies global
uncertainty Not locally accurate
Makes avaiable multiple realizations
Facies Simulation
Facies are considered an indicator (categorical) variable and simulating facies requires indicator simulation.
Lecture 6: Geostatistical Mapping Concepts, The Quiz