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  1. First Order Error Propagation (FOEP)

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- [Stein] The presentation today is based on my paper and also my presentation that was given at SPWLA last year in Iceland, and the plan today, first, go through the subject that was presented at Iceland last year, that will take close to twenty minutes and after that we'll spend ten minutes to show one example that illustrates one application of the presented workflow. Then we start. As already mentioned, the title on this presentation is on overcalculation and use of first order. Overpropagation is an integral part of petrophysical calculation. The back one for this work is my view that calculation of error is an essential part of measurement and scientific calculations and should also be an essential part of petrophysical calculations. If not, our petrophysical results have lack of confidence. How the error is calculated is not that important, but as an integral part of the computer script that arrive the petrophysical results. First order of propagation is the preferred method. The goal with this presentation is to explain why. This presentation includes first introduction to error calculation and error propagation in petrophysics, then the first order error propagation method will be explained. I will show some examples where first order error propogration is applied as an integral part of the petrophysical calculations. And finally, the summary of this presentation. These four pointers is the subject that was presented in Iceland. After that, as mentioned, I will go through an example that illustrates one application of this presented workflow. This work is based on the following view. Error is the difference between the measured or calculated value and the true value. The petrophysical estimators are more or less empirical based equations. Empiricism is the theory of knowledge. Lack of knowledge is part of the error. The error must reflect the level of knowledge as well as the data quality. The evaluation of the error is a natural part in the process in order to choose the model and input to the model, and should be quantified from the very beginning in the petrophysical workflow. To obtain confidence in the petrophysical results, petrophysical results should include quantified and traceable errors. And the last point here is the main subject for this presentation, and that is errors can be an integral part of the petrophysical calculations by using first order of propagation as an integral part of the computer script that calculates the petrophysical results. Error propagation. Error propagation is the method of determining an uncertainty in a function of independent variables each with an uncertainty, and is the problem of figuring out how the integral interrelated propagated throughof estimates. And here exists a couple of techniques for doing that. And here we see the most common used technique in the petrophysical communities. Sensitivity analysis, at some incidents, a specific uncertainty is highlighted as the dominant one and in such a case we can do insensitivities on these or that We have the well known Monte Carlo simulation method which is a statistical approach. And we have also the well known first order error propagation method which is an analytical approach. There isand there exists different views on the different methods. My view is that all techniques are valuable as long as the values on the model inputs, the values in the errors and the error propagations can we explained and traced. But as an integral part of the computer script that arrived at petrophysical results, first order error propagation are, for a couple of reasons I will come back to in the summary, the preferred method. One of the reasons we can see from this equation. We can see that the equation and the math is quite simple. It consists of only a vector C and a matrix sigma. C and sigma will be explained. C is the vector that contains the constant in a linear multivariate equation. Sigma is the matrix that contains the variance to the variables and the covariance to the variables. From basic statistics we know that the variance to the function, which is the square arrow, can be described by the sum of the variance and the covariance or by applying the vector notation like this. Note that this equation is only valid for a linear multivariate function. In case of unknown linear equations, for instance, arches equation, we have to do a linearization. This linearization can be performed by applying a Taylor series expansion on the estimate as shown by this equation. This is a standard mathematical way to do a linearization and I will not go into detail for that equation here. The whole point with this slide is to display the following three points. A linearization can be performed by applying a Taylor series expansion on the estimate. This relatively complex expression can be simplified to a first order approximation if delta X, the value of the uncertainty, is small compared with X, the value in the If then so is, our expression for error is still approximately valid if we replace the constant in the C vector with partial derivative to the equation. That's all. In summary, in summary, the whole first order error propagation I've written consists of only three simple steps. The first is to define the C vector, which is the container that contains the partial derivative to the equation. Second is to define the sigma matrix which includes the variance at the diagonal and the covariance to the variable of the diagonal. And finally, calculate the error, which is simple. These three points can very simply be precoded into the computer as an integral part of our petrophysical estimates. This is done once for each of our estimates. The petrophysics is to feed the sigma with value. It is to find the value and uncertainty in all the integer to the equation, the delta X, and to define the covariance between the variables, which is given by the correlation between the variables.