A Mathematical Analysis of Errors in Directional Survey Calculations
Authors: Harvey R.P. , Walstrom J.E. , Eddy H.D. 
Published: December 2007

This is a summary and an extension of the ideas presented in two earlier papers dealing with the effects that data errors and model errors have upon the results from directional survey calculations. An important conclusion is that if any of five specific models is used the uncertainty, due to errors in data is much more serious than the uncertainty due to errors caused by the model.

Introduction
The degree of uncertainty in our knowledge of a position in a wellbore is important when dealing position in a wellbore is important when dealing with highly deviated wellbores drilled from urban, Arctic, and offshore drillsites. In this paper we shall draw heavily from the ideas developed in two earlier published articles, which should be read in conjunction with this paper for a complete understanding. In those papers we suggested that any errors in the calculation of a position in a wellbore can be analyzed as arising position in a wellbore can be analyzed as arising from two sources. One source of error is the inaccuracies in the data readings due mainly to instrumentation error. The other, usually lesser error. which we call model error, is due to the inaccuracies introduced by the use of an approximating model. Of course there are other obvious sources of error-common to any calculation - such as errors introduced when transcribing numbers, arithmetic errors, and errors due to the use of finite precision arithmetic either on or off a computer. However, we will assume that with care any errors in this category can be eliminated or made negligible. The errors arising from the two principal sources are discussed in the next two sections, which are then followed by a section dealing with the composite error. In our notation, probability distribution parameters pertaining to errors due to data inaccuracies are primed, parameters pertaining to model errors and to composite errors are pertaining to model errors and to composite errors are double primed and unprimed, respectively. We will describe the analysis as though the error were for the bottom-hole point, but bear in mind that the ideas can be applied to point, but bear in mind that the ideas can be applied to any other point in a wellbore. Following this, the inverse problem is considered: "How should a directional survey, and perhaps the well course, be planned so that the errors are kept within prescribed bounds?" prescribed bounds?" Errors Due to Data Inaccuracies
In Ref. 1 we studied the effect of data reading errors on the calculated bottom-hole coordinates. Joint error characteristics for the survey reading instruments were estimated through discussions with representatives of the instrument manufacturers and service organizations. These characteristics in the form of error probability distributions are converted by a statistical analysis that produces the joint error probability distribution for the bottom-hole coordinates. The details of such a study are presented in Ref. 1. which could be applied to the Balanced Tangential Method." The analysis required in any particular case would of course depend on the model used. We have not performed the analysis for other models in general. It may not be possible to carry out the integrations analytically for any particular model, but in any case the Monte Carlo approach could be used to obtain the joint error distribution for results of interest by random sampling and simulation. Results are interpreted statistically as follows.
JPT
P. 1368