First Section

- In this lecture we will be talking about kriging, which is very classical, geostatistical interpolation technique. The name kriging actually has been given by Matheron, one of the founders of geostatistics and named after Danie Krige who was a South African geostatistician who worked in the mines of South Africa, and really was the first person to recognize spatial continuity and its effect on estimation. Considering the first slide here a typical situation that kriging tries to address. Which is we have some data here four sample data and we have some estimation to be estimated which is the orange location. The simple question is: what is the best guess of the unknown property at the orange location given the four data available? In kriging, we will be writing this estimator as a linear combination of data and provide weights to each sample data. So the question is then: what are these weights? There are two principles that can be used here. The first principle of course is that data that are closer in some geological sense we'll see in a bit, will have to get bigger weights but also that data that are close together such as these two samples at the top right here should share some of the weight because they are redundant. Kriging has many interesting properties that differentiate it from the classic interpolators such as line interpolation and other interpolation methods and we'll be covering some of those. So what is kriging then? As I mentioned before, kriging is a spatial estimator or interpolator, it handles spatial or spatial-temporal data. It also provides a single best guess. In other words, it doesn't provide an uncertainty it simply provides a single number. As you notice it's a linear estimator in the sense that it combines data linearly. It's also the best possible linear estimator that is also unbiased. This means that if we would repeat kriging over many situations then we'll find that the true value is on average, equal to the estimated values. And one of the more important properties of kriging is that it accounts for spatial correlation as measured by the variogram. And for variogram I'd like to refer to that particular lecture. Consider first a simple example. On the left we have some true unknown field for example, log-permeability or porosity and we have from the true field only very few sample data available. The idea of kriging is to provide an interpolation of this data that also matches this data interpolates this data exactly and on the right hand side we see the resulting kriging surface. You will notice immediately that the kriging surface is much smoother than the true phenomenon and we'll come back to this concept. How does kriging achieve this? The method's actually quite complicated and we'll just see and provide a summary of what the end result is. At every location we have determined a set of weights. In this particular case we notice how the weights are determined by a linear system of equation with some constraints. What's important about this equation is this C here, which is the covariance, which can be related to the variogram. In other words in order to perform kriging we need to know the variogram of our data. And so this is the fundamental assumption or the fundamental information that needs to provide through kriging. On the bottom you also notice that kriging provides what is called the kriging variance. The kriging variance is an estimation variance which essentially states how confident are we about this single best guess. And for this estimation variance there is also some closed-form expression. Kriging comes with a number of flavors. And these flavors can be understood in how we understand that our formula is spatially distributed. And this is the equation of the top right corner. The top right corner we write that our phenomenon for example porosity, which in this case is the variable Z, at various location x can be decomposed into two parts. The first part is some unknown mean part a mean variation of the porosity. And the second part is a residual part, which says that on top of this mean variation there is in addition a higher varying residual component. Dependent on what we say about the mean, we have essentially a number of flavors of kriging. For example in simple kriging, we essentially assume that the mean is constant and no, that means there's not a function of x, it's no longer the entire domain. This evidently is a very strong assumption of stationarity of the mean. In ordinary kriging we get somewhat rid of this assumption, while assuming that the mean is not known and sort of filtered out of the equation but we sort of still assume it to be somewhat constant. If we really have an explicit information about what tihs mean is we can do kriging with locally varying mean. For example I can use seismic data and I can turn the seismic data and say that the seismic data is a mean porosity by some form of regression or calibration. And then kriging allows me to constrain to the well data. In its most general form, kriging is also known as universal kriging where we don't necessarily make a lot of assumption about the mean except that it is some function of trend information or a drift information and I'll show a short example in the next slides. Here we have a simple example of what is universal kriging also known as trend kriging. On the left we have some reference field it looks quite complicated in fact it's a mountain range close to the California-Nevada border. And we assume that this mountain range is largely unknown and what we have is some data at certain locations. So in reservoir context this could consist of well data at certain locations. We also assume we have some information about the major trends that happen in this particular field which in the reservoir case can be extracted from seismic data. And this seismic data is provided imagine here on the right hand side. In universal kriging as I mentioned before we need to have the variogram of this particular field of the 100 sample data. And this variogram is shown on the left. Actually it's not the variogram of Z itself, it's the variogram of the residual which is obtained if you consult some more theoretical books on this is obtained by subtracting the mean function. Given this variogram we can model the variogram and we could use this variogram in the kriging system and obtain a kriging estimate. With the 100 data points we notice that the obtained estimate is very different from the original mountain range. And this shows that kriging is indeed a smooth interpolator. However as the data increases, the kriging becomes much more heterogeneous and the fields obtained as such are much more heterogeneous and so approaches a more realistic mountain range. What are properties of kriging? These properties are in fact important if you would like to consider other interpolators that are not inspired by theory of random functions. One interesting property of kriging is that it's an exact interpolator that means it will exactly match the data at the locations where data is available. This is not common with most other interpolators such as line or other interpolators. It also provides a sense of error of the interpolator which also does not come with traditional interpolators and this error is expressed by the estimation variance or kriging variance. If some secondary or other information is available then this can be accounted for by some drift or trend turn that is incorporated into the mean. It can also account for measurement error which is error on the sample data and this is often done through the nugget effect. However as most interpolators it returns a map that is much more smother than the true variation. And this is important to understand because for that reason we cannot use kriging to create permeability maps where the actual true variation is very important. Therefore in summary we state that kriging is an interpolator that is statistically inspired as opposed to other interpolators. This comes with a number of nice interesting properties such as exactitude, the existence of estimation variance, and the use of variograms. This is also a requirement for kriging. There has to be a knowledge of the variograms. We discussed how various flavors of kriging exist which allow informing the mean-variation in different ways depending on the case we have and that does require some subjective decision.