SPWLA 46th Annual Logging Symposium, June 26-29, 2005

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Error Properties of Magnetic Directional Surveying Data

Erik Nyrnes, NTNU–Norwegian University of Science and Technology, Torgeir Torkildsen, Statoil ASA Norway, Hossein Nahavandchi, NTNU

Copyright 2005, held jointly by the Society of Petrophysicists and Well Log Analysts (SPWLA) and the submitting authors. This paper was prepared for presentation at the SPWLA 46th Annual Logging Symposium held in New Orleans, Louisiana, United States, June 26-29, 2005. ABSTRACT The primary tools for directional surveying while drilling (MWD) are based on the determination of the magnetic azimuth. This paper presents the results from a study that highlights several aspects of the quality of directional surveying with magnetic tools. Survey data from 66 North Sea well sections have been investigated in order to evaluate the overall quality of the sensor readings and the applicability of different estimation and error detection techniques. The surveys cover different wellbore geometries and are performed by several service companies. The Industry Steering Committee on Wellbore Survey Accuracy (ISCWSA) has developed error models for magnetic directional surveys, which now have become an industry standard. These error models are derived for two standard single-station processing techniques. The error term values are settled with the basis in input from several service companies. The results from the analysis of the survey data presented in this paper confirm the systematic error term values in general. However, some of the error term accuracies are normally better than the modelled ones. The applicability of multi-station estimation techniques is demonstrated. There is always a chance of misinterpretation due to several reasons. These are: poor geometry, low redundancy, high random noise level, presence of gross errors, and errors in the Earth's gravity and magnetic field references. Since the estimated results are very dependent on the selected parameter model, the major issues related to the selection procedure are discussed. Procedures are also derived for performing multi-station estimation of parameters in a reliable and robust manner. Finally, there is a practical demonstration of the application of a new error detection procedure. This method is used to detect outliers that might cause errors

in the multi-station estimation. This is a significant potential source of error as such outliers cannot be detected when applying conventional methods. It is demonstrated how much these extended quality control procedures can improve the final wellbore positions. Key words: Wellbore positioning, Directional surveying, Quality control, Multi-station estimation techniques, Systematic errors, Gross errors, Random errors.

INTRODUCTION Measurements collected from a number of survey stations can through a least squares adjustment be used to detect and account for different error terms on all measurements, see [Brooks et al. 1998] for details. Such techniques are in the following denoted Multi-Station Estimation (MSE). In this paper the Gauss-Newton method is used to estimate the parameters, see the Appendix B. The different error terms are commonly divided into systematic, random and gross errors. Systematic and gross errors are in this study treated as unknown fixed parameters, the only difference being the assumptions about whether they affect a number of measurements (systematic) or single measurements (gross). They will be estimated together with the directional parameters for each survey station; the magnetic azimuth Am, inclination I and high side toolface τ. The estimations are based on the linear regression model:

[ ] ee2

ee)Cov( ; )(E QΣyβ

ZXeyy σ==⎥⎦

⎤⎢⎣

⎡∇

=−= (1)

where y is a vector of n observations, β is a vector of u unknown fixed parameters Am, I and τ, X is a n×u known coefficient matrix, Z is a n×r known coefficient matrix corresponding to the r additional model parameters ∇, Qee is a known observation error cofactor matrix and σ2 is the variance of unit weight, usually unknown. The inverse of Qee is the weight matrix, 1

ee−= QP .

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The additional parameters ∇ may represent the systematic error terms; linear scale factor ν, bias η and sensor misalignment α, or gross errors (biases) in single sensor readings (denoted ∇i). ∇ may also represent the Earth's reference components; gravity G, field strength B and dip angle θ (defined in Appendix A). The errors e are in this context assumed to be normally distributed, e ∼ N(0, Σee) and uncorrelated (Σee diagonal). This quantity will often be referred to as noise. The scale factor ν, bias η and the errors e may also be attributed to environmental effects, and not necessarily sensor imperfections only. The aim of this study is to evaluate the overall quality of MWD magnetic directional surveys with respect to all the three error categories described above. 66 survey-sections have been analysed in detail. The surveys are performed by several survey companies in the period between 1998 and 2004 and cover the northern parts of the North Sea and the southern parts of the Norwegian Ocean. They represent various geometries and borehole dimensions with average lengths of roughly 30 stations. The spread of the high side toolface is considered good for all wellsections. First, the properties of the systematic error terms are investigated. The estimated error terms will be compared to the error term values of the ISCWSA MWD-error model, which consider the lumped effects of the bias and linear scales, and two types of sensor misalignments, see [Williamson 2000] for details. Therefore, only the bias and scale error terms are considered in this study. The corresponding error values are listed in Table 8. (There is one exception, see the Example 2, page 4). Some major issues related to the selection of model parameters and misinterpretation of the estimation results are discussed, in order to provide a better understanding of multi-station estimation techniques. Second, the frequency of gross errors in the single sensor readings is investigated. As undetected gross errors may affect the results considerably, the process of removing them is important and will therefore be discussed in detail. See also [Nyrnes et al. 2005]. Finally, the noise levels in the measurements are evaluated. The ability to detect systematic and gross errors in the measurements is dependent on this quantity. For the magnetometer readings the estimated noise levels are compared with the expected time and location dependent standard deviations of the magnetic field intensity variation to see if there are any agreements.

METHODOLOGY Test of hypothesis. The general test to decide whether the subset ∇ of systematic error terms is zero is based on the following statistical hypothesis:

[ ] 0 , )(E :H vs. )(E :H A0 ≠∇⎥⎦

⎤⎢⎣

⎡∇

==β

ZXyXβy

(2)

where H0 is the null hypothesis (∇ is zero) and HA is the alternative hypothesis (∇ is different from zero). H0 is rejected if the 100(1−α) % confidence region for ∇ does not contain 0, i.e. if:

[ ] [ ])(F

ˆr

0ˆ 0ˆT rqn,r2

Trˆˆr

rr α>σ

−∇−∇= −−

∇∇Q (3)

where

rrˆˆ ∇∇Q is the joint cofactor matrix of the esti-

mates in the subset, 2σ is the σ2 estimate under HA and Fr,n−q−r(α) is the upper (100α)th percentile of the Fisher distribution with r and n−q−r degrees of freedom. In this study the significance level α = 0.05 is used for the test of the null hypothesis H0. Selection of specific model parameters. Since the goal is to distinguish specific model parameters of importance, the simultaneous confidence property in Equation (2) is ignored. Instead, important parameters will be detected according to a stepwise regression approach similar to the so-called Backward Elimination explained in [Miller 2002]. In a complete model one starts by deleting the parameter corresponding to the smallest residual sum of squares after deletion (which is the one with the smallest Student's t-test statistic). Then a new adjustment is performed and the process is repeated until only significant model parameters are left. It should be considered that even the resulting subset fits the measurements y fairly well, it may just be one among several model combinations that have similar effects on the residual sum of squares Ω. Derivation of measurement-weights. The Earth’s magnetic field is commonly defined by the field strength B, magnetic dip angle θ and the declination δ, see Appendix A for details. The irregular variations in the strength and direction of the Earth’s magnetic field vector will give rise to irregular variations in the measured magnetic field strength along all three instrument axes. The assumptions of similar uncertainties for the errors e in all three directions are therefore considered reasonable. See also [Torkildsen et al. 1997]. The errors in the accelerometer measurements are also assumed to have the same

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uncertainty for all three axes. The coordinate systems are defined in Appendix A. The following approach is used to find an approximate weight relation between the accelerometer and magnetometer readings. The variance of unit weight σ2 is first estimated from the accelerometer measurements only. Then the magnetometers are introduced with corresponding weights such that the original variance of unit weight estimate 2σ is left unchanged. The Earth's magnetic field components are introduced as measurements with a priori uncertainties depending on whether standard referencing (IGRF or BGGM) is applied solely or together with corrections for the local crustal field (enhanced referencing). See Table 1 for details.

Reference quality σB σθ Standard Referencing 130 nT 0.20 degEnhanced Referencing 65 nT 0.12 deg

Table 1: Standard deviations for the standard referencing according to [Williamson 2000] and Statoil values for the enhanced referencing. The prediction of the local gravity is based on the international gravity formula and corrections for local anomalies. See Appendix D for details. The gravity values provided by the survey companies are used if the measurements were given in other units than ms-2 (e.g. counts). The measurements of the Earth’s reference components are introduced as average values for the entire section. Interpretation of the results. The interpretation of the estimation results is not straightforward because we often have to choose between several parameter models which have similar effects on the measurements y. As an example, consider the subset consisting of the x, y, and z-magnetometer scale errors and the Earth's magnetic field strength B. The columns in the design matrix Z representing these parameters are linear dependent irrespective of the geometry and hence they cannot be estimated simultaneously. From standard theory this singular problem is often referred to as exact collinearity. An error in the a priori Earth's magnetic field strength B will induce equal errors in all three magnetometer scale estimates. Accurate a priori information about B is therefore needed in order to estimate all scales properly. Let us suppose that B is estimated (i.e. no a priori information is used) and that the axial scale error is not

estimated. Suppose also that the true field strength B is constant for the whole section. If the true axial scale actually is positive and of significant order of magnitude, it will cause the cross-axial scale estimates to decrease equally and cause the field strength estimate to increase. Other estimates are left unaffected. The a priori field intensity B may then be suspected to be of poor quality, since it is significantly larger than the estimated value for the entire well, and the drilling fluid may be suspected to contain magnetically susceptible materials since the transverse scale estimates are negative (indicating attenuation of the field intensity) and of similar order of magnitude. Let us consider another example. Suppose that all three scales are to be estimated and that the a priori field strength B is introduced as a measured quantity. In addition, suppose that the magnetic field intensity observed by the sensors actually is damped. If B is introduced with a lower value than the true value, which by coincidence is sufficiently low to force the cross-axial scale estimates to zero (or close enough to zero to be declared insignificant) this may lead to the conclusion that a positive axial scale error is the probable reason for the model misspecification. For the accelerometers the true error terms must be even larger to be detected than for the magnetometers, although the accelerometer measurements in general are less noisy. The main reason is that the redundancy is lower than for the magnetometers, maximum one on average at each station. The introduction of magnetometer readings will be helpful, but not always of considerable importance. In the same way as for the magnetometers, the estimation of the accelerometer scale errors is sensitive to errors in the local gravity. For straight wellbores, in addition to the cross-axial biases it is only possible to estimate two of the four remaining accelerometer error terms. This gives high odds that undetected effects may remain and the chance for misinterpretation is therefore high. As a general rule, the estimation of the four transverse accelerometer and magnetometer bias error terms (ηgx, ηgy, ηbx, ηby) can always be considered highly reliable. Among all the additional parameters, the cross-axial bias estimates are the ones which are least sensitive to errors in the sensor readings, reference components and unmodelled effects.

EXAMPLES Four datasets with special error characteristics will now be analysed and discussed in detail.

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Example 1. As will be demonstrated for the horizontal wellbore shown in Figure 1, the results of the MSE are very dependent on the selected parameter model. Note that the azimuth varies a lot in the first half of the wellbore, and remains almost constant in the second half. It can be seen that the parameter models defined in Table 2 lead to considerably different azimuth estimates, especially in attitudes close to the horizontal-east direction. Model 1 is considered to be the most trusted solution and is therefore used as reference.

Model 1 2 3 4 5 6 7 All biases est est est est est est est x & y-scales fixed est est est est est est Axial scale fixed fixed est fixed fixed fixed est

B meas meas meas est meas est fixedθ meas meas meas meas fixed fixed fixed

Table 2: The parameter models applied for the Example 1. Est = estimated, meas = measured, fixed = not estimated (reference fixed to initial value and systematic errors fixed to zero )

Azimuth differences. Multi-station estimation.

Station no.(Interval: 30 m.)

0 10 20 30 40

Azi

mut

h di

ffer

ence

s (de

g.)

0.0

0.5

1.0

1.5

2.0

1234567

Wellbore geometry

0 10 20 30 40

Deg

rees

30

40

50

60

70

80

90

100

IncAz

Station no.

Parameter model

Reference: Model 1

Figure 1: Example 1. Magnetic azimuth estimates for the parameter models defined in Table 2. Some of the models produce azimuth estimates differing almost two degrees in the east direction. It can be seen that Models 6 and 7 provide the largest azimuth differences. The estimates are very sensitive to errors in the reference components and sensor readings in directions near the horizontal east-west. They produce similar results as noted earlier. Further down the well the azimuth differences form two main trends depending on whether the magnetic dip angle is measured or fixed to the initial value (i.e. treated as an error-free quantity). The difference is approximately 0.5

degrees in parts with constant inclination and azimuth. Model 3, which is a realistic one, produces considerably more unstable azimuth estimates than Model 2 in directions close to horizontal east-west. This is due to the introduction of the axial scale error in the model. Example 2. This section turns smoothly from the east to north direction and the inclination is roughly 40 degrees for the whole section, see Figure 2. Large scale errors were detected on the transverse axes, probably caused by magnetic particles in the drilling fluid. It can be seen that the gyro survey verifies the correctness of the MSE results. The left hand plot in Figure 2 shows that the azimuth differences between the SSE and MSE are largest for the east direction, and almost zero for the south direction. This is in accordance with the propagation of transverse scale errors of equal magnitude, since the joint effect is zero for the south direction. The right hand plot in Figure 2 compares the predicted magnetic field strength with the field strength calculated from the measurements. The differences are largest for the south direction and increase drastically as the inclination becomes greater in the final parts of the well. Even if the scale errors are very large, they cannot be detected using conventional methods (i.e. comparing the measured field strength with values predicted from independent sources) in attitudes where the effect on the calculated field intensity is small. The worst case is when drilling closely parallel to the Earth's magnetic field vector.

Station no.0 10 20 30 40

Azi

mut

h di

ffere

nces

(deg

.)

-2

-1

0

1

2

3

4

5

MSE (reference)SSEGyro

Station no.

0 10 20 30 40

Mag

netic

fiel

d st

reng

th (n

T.)

50800

51000

51200

51400

51600

51800

52000

B (nominal value)B (calculated)

Wellbore Geometry

Station no.

0 10 20 30 40 50

Deg

rees

0306090

120150180210

InclinationAzimuth

Station spacing: circa 30 m.

Figure 2: Example 2. Left hand plot: The simultaneous effects of the transverse scale errors on the azimuth estimates are almost zero for the ordinary SSE (Single-Station Estimation without axial correction) technique in the east direction. Right hand plot: Comparison between the calculated and predicted magnetic field intensity.

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Example 3. MSE was first applied with a model consisting of biases and scale error terms. A large axial magnetometer bias was detected. After a comparison with a gyro-survey, the azimuth deviations were still up to four degrees, see Figure 3. After a more detailed analysis, the most likely reason for the azimuth differences was found to be an axial misalignment between the accelerometer and magnetometer sensor packages in the survey tool. The estimated misalignment was 1.2 degrees and strongly significant. The improved model is used as the reference in Figure 3. It can be seen that the azimuth differences are largest in the most inclined parts of the wellbore, and somewhat smaller in the least inclined parts. This is in accordance with the propagation of a toolface error [Ekseth 1998]:

τΘ

Θ Θ= d

cosIcoscos-AcosIsinsindA (4)

At an inclination of 80 degrees, Equation (4) shows that a toolface error of 1 degree induces a negative error in the azimuth estimate of about 3.5 degrees. The MSE (with misalignment parameters) and SSE with axial magnetic correction lead to a wellbore position difference of approximately 170 metres.

Station no.

0 20 40 60 80

Azi

mut

h di

ffere

nces

(deg

.)

-1

0

1

2

3

4

5

Optimal model (Reference)Misalignment parametersnot modelledGyroscopic azimuth

Well bore geometry

Station no.0 10 20 30 40 50 60 70 80 90

Deg

rees

0306090

120150180210

InclinationAzimuth

Station spacing: circa 30 m.

Figure 3: Example 3. Comparison between the gyroscopic azimuths and the MSE-azimuths with and without sensor misalignment parameters in the model. The azimuths of the improved model (including misalignment parameters) agree most with the gyro azimuths. Example 4. The first part of this horizontal wellbore lies near the east-west direction. The results produced by three different estimation techniques (MSE, SSE with and without axial correction) are compared in Figure 4. Because the axial magnetometer bias is very small and insignificant, the MSE and SSE without magnetic correction produce similar azimuth estimates. The SSE with axial correction produces very unstable azimuth estimates in the horizontal east direction. The

differences are larger than 5 degrees compared with the MSE and SSE (Single-station estimation without axial correction), see Figure 4. The application of the axial magnetic correction technique leads to a 30 metres displacement in position for this section, compared with SSE without axial correction.

Survey station no.(Interval: circa 30 m.)

0 2 4 6 8 10 12 14

Azi

mut

h (d

eg.)

50

60

70

80

90

Azimuth (standard)Azimuth (MSE)Azimuth (SSE ax. mag. corr.)

Wellbore Geometry

Station no.0 10 20 30 40

Deg

rees

020406080

100120

InclinationAzimuth

Figure 4: Example 4. The application of the axial magnetic correction technique leads to erroneous azimuth estimates near the east-horizontal directions.

SYSTEMATIC ACCELEROMETER ERRORS Cross-axial bias errors. Totally 41 x-axial and 44 y-axial biases were declared significantly different from zero by testing the hypothesis defined in the Equations (2) and (3). The corresponding cross-axial bias estimates are shown in Figure 5. The largest estimates are about ±1 Gal in magnitude.

Significant x-axial accelerometer biases (Gal)

Estimates (Gal)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Freq

uenc

y

02468

101214161820

Significant y-axial accelerometer biases (Gal)

Estimates (Gal)

-1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6

Freq

uenc

y

02468

101214161820

Significant cross-axial accelerometer biases (Gal)

Estimates (Gal)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Freq

uenc

y

0

5

10

15

20

25

30

Figure 5: Estimated values of significant accelerometer bias error terms.

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Nearly 50 % (19 out of 41) of the x-axial biases have their values outside the interval ±0.4 Gal, which are almost 30 % out of the 66 examined sections. As a comparison, the corresponding value of the MWD-error model is approximately 0.39 Gal (1 standard deviation). For the y-axis bias estimates, about 40 % (18 out of 44) of the absolute values exceed this level, equivalent to about 30 % out of all the examined survey sections. The x and y-axial bias estimates show some different properties. While the x-axial bias estimates are symmetrically distributed around zero, we see that a large number of the y-axial biases are negative. Although the parameters according to the tests are classified as gross errors, most of them are hardly large enough to have any practical influence on the final wellbore positions compared with other sources of systematic errors. Due to the normality assumption of the measurement errors and the MWD-error model terms, maximum one third of the estimated error values should lie outside the interval ±0.39 Gal to be in accordance with the MWD-error model. We see that this criterion is fulfilled for both axes. The surveys are considered to be of acceptable quality with respect to the cross-axial bias error terms. For about 50 % of the wellsections (34 out of 66), none of the cross-axial bias estimates turned out to be larger than 0.39 Gal in absolute value, regardless of whether the parameters differed significantly from zero or not. The ability to detect the cross-axial biases is usually very good, even if they are considerably smaller than the corresponding MWD-error model values. Axial bias errors. The variance of the estimated axial bias usually becomes high, especially in the presence of the axial scale in the model. The effect of an undetected axial scale propagates mainly into the axial bias estimate. The axial bias estimate is sensitive to errors in the gravity reference G. Due to these uncertain factors, the axial bias estimate can usually not be trusted. Scale factor errors. A proper estimation of these parameters requires precise knowledge about the local gravity. However, the estimation involves several other uncertain factors, as will be explained below. The variances of the cross-axial scale estimates usually become similar and much lower than for the axial one. If the estimates themselves also turn out to be similar, the cause may be a gross error in the a priori gravity, and not necessarily sensor imperfections. Because the variance of the axial scale estimate usually becomes high, the effect of the sensor reading errors usually

masks the effect of the gravity error. Like the gravity error, an undetected axial scale (i.e. one that is not estimated) also affects the cross-axial scales equally. The ability to detect the cross-axial scale errors is reasonable good, provided that the gravity reference is of good quality. Usually, errors even smaller than the corresponding MWD-error model value (0.05 %) can be detected with high probability. Sometimes the a priori gravity reference was suspected to be of poor quality, since the cross-axial estimates became similar and strongly significant (also in the presence of the axial scale in the model.) However, it was considered reasonable to estimate the cross-axial scales for several bit-runs. RMS values for the estimates of significant error terms turned out to be almost 0.1 % for both axes (see Table 8). However, when considering the potential disagreement between the predicted and true gravity, there is no reason to claim that the true errors are larger than specified by the MWD-error model. Since the differences between the x and y-axial biases for each pair of estimates also tended to be small (not shown), this assertion is further confirmed.

SYSTEMATIC MAGNETOMETER ERRORS Cross-axial bias errors. The two upper histograms in Figure 6 show that the cross-axial bias estimates are similarly distributed and almost of the same order of magnitude. All estimates of significant error terms have their values within the interval ±200 nT except of a few x-axial biases of almost −250 nT in magnitude (see the histogram to the left). The majority of estimates are smaller than 100 nT in absolute value for both axes. The cross-axial estimates are represented together by the histogram in the middle of Figure 6. We see that the largest number of estimates, approximately 70 % out of the 58 significant, have their values inside the interval of tolerance: ±70 nT. This means that only about 15 % out of the 132 (2×66) potential estimates exceed the limits ±70 nT.

The results indicate that the true errors generally are lower than that modelled by the MWD-error model. Estimates exceeding the tolerance limits only corresponded to much less than one third of the surveys. For about 80 % of the examined sections, none of the cross-axial bias estimates turned out to exceed the limits ±70 nT simultaneously for the same section, regardless of whether they differed significantly from zero or not.

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Even if the cross-axial bias errors are far below ±70 nT, they can still be detected with high probability. The overall ability to detect these error terms is considered good.

Estimated x-axial magnetometerbiases (significant).

Estimates (nT)

-200 -100 0 100

Freq

uenc

y

0123456789

10

Estimated y-axial magnetometer biases (significant).

Estimates (nT)

-200 -100 0 100 200

Freq

uenc

y

0123456789

10

Estimated cross-axial magnetometer biases (significant).

Estimates (nT)

-200 -100 0 100

Freq

uenc

y

0

2

4

6

8

10

12

14

Estimated z-axial magnetometer biases, less than 1000 nT and larger than -1000 nT (significant).

Estimates (nT)

-750 -500 -250 0 250 500 750

Freq

uenc

y

0

2

4

6

8

10

Estimated z-axial magnetometer biases, larger than 1000 nT and smaller

than -1000 nT (significant).

Estimates (nT)

-4000-2000 0 2000 4000 6000 8000 10000

Freq

uenc

y

0

2

4

6

8

10

Figure 6: Estimated magnetometer bias error terms. Axial bias errors. We now consider the axial bias estimates. Estimates of smaller magnitude (within ±1000 nT) are shown on the lower left histogram in Figure 6, while the larger ones (outside ±1000 nT) are shown on the lower right. The corresponding ISCWSA tolerance is set equal to the lumped uncertainties of the systematic sensor reading error (70 nT) and the axial magnetic interference (150 nT); derived from the AMID parameter in [Williamson 2000] for the North Sea conditions:

nT 165nT) (150)nT 70( 22 ≈+ (5)

If the axial magnetic interference is unacceptably high, an axial magnetic correction technique should be

applied if the conditions permit, i.e. if the deviation from horizontal east-west is acceptable. By doing so it does not matter theoretically how large the true bias is. However, an axial error will have the largest influence on the azimuth estimation in such positions if standard SSE is applied. About 35 % of the sections were surveyed without correcting for axial magnetic interference. Significant biases were detected for the majority of these surveys. Most of them were some hundreds of nanoteslas larger than the tolerance (165 nT) and some almost ten times larger. Some basic statistical parameters are given in Table 3. The results show the potential magnitude of the axial bias errors, which mainly reflect the magnetic interference from drillstring/BHA.

Mean Std.dev. Max. Min. 470 nT 370 nT 1548 nT 202 nT

Table 3: Basic statistics of the absolute values of the axial bias estimates for surveys which had not been corrected for axial magnetic interference. Cross-axial scale errors. If either the axial scale or the Earth's magnetic field strength B deviate significantly from its initial values, the values of the cross-axial scale estimates are dependent on whether we choose to estimate the axial scale or B. This is shown in Figure 7 and will be discussed more in detail in the next section. We see from the upper plots in Figure 7 that some of the estimates are more than 2 % in magnitude. One is almost 4 % (geomagnetic references measured). The smallest error is approximately 0.25 %, which is almost 0.1 % larger than modelled (0.16 %). In general, errors of such small magnitudes can hardly ever be detected since the magnetometer measurements usually are too noisy. This is mainly due to the high level of irregular variations in the Earth's magnetic field strength. As a result, the transverse scale errors turned out to be insignificant for about 60 % of the examined surveys sections. The cross-axial scale estimates are plotted against each other in Figure 7. We see that all estimates (except from the one in the right plot) are negative and of similar order of magnitude. It is well known that the magnetically susceptible properties of drilling fluid (due to steel particles from wear, weight material etc.) may lead to a cross-axial damping of the Earth's magnetic field intensity surrounding the sensors and give rise to cross-axial scale errors of similar magnitude [Wilson, Brooks 2001], [Torkildsen et al. 2004].

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In this study the errors are defined in such a way that they will become negative in the case of damping. The results of the analyses presented in this paper show that the damping is larger when ilmenite is used compared with barite. The attenuation also tends to increase for greater dimensions and is often larger for the 12.25 inch sections than for the 17.5 inch sections. This was also pointed out in [Torkildsen et al. 2004].

Estimated x-axial scale factor errors (significant).

Estimates (%)

-4 -3 -2 -1 0

Freq

uenc

y

0

2

4

6

8

10

12

Axial scale errors estimated,if significant.

Axial scale errors not estimated.

Estimated x-axial scale factor errors (significant).

Estimates (%)

-4 -3 -2 -1 0

Freq

uenc

y

0

2

4

6

8

10

12

Estimated cross-axial magnetometerscale factor errors (significant).

Estimated x-axial scale errors (%)

-5 -4 -3 -2 -1 0 1

Estim

ated

y-a

xial

scal

e er

rors

(%)

-5

-4

-3

-2

-1

0

1

Estimated cross-axial magnetometerscale factor errors (significant).

Estimated x-axial scale errors (%)

-5 -4 -3 -2 -1 0 1

Estim

ated

y-a

xial

scal

e er

rors

(%)

-5

-4

-3

-2

-1

0

1

Figure 7: Estimates of cross-axial magnetometer scale error terms deemed significant according to test. Left hand plots: The axial scale error is not estimated and the geomagnetic reference is estimated, if it is significantly different from initial value. Right hand plots: The axial scale error is estimated if it is significant, and the geomagnetic reference is measured. It should be noted that large scale errors of similar magnitude can also be attributed to sensor imperfections, if by mistake the input magnetic field intensity B is given a wrong value in the calibration process. If the a priori magnetic field intensity deviates significantly from the true average value of the entire section, it will give rise to similar scale errors on all three axes. For the magnetic dip angle, an equivalent error will affect the cross-axial scales equally, but the axial scale differently. Since the error characteristics were similar for about 40 % of the sections, there is a fair reason to believe that magnetic environmental effects are the main cause for all of these surveys. When considering the sensor specific errors only, it is unlikely that the true scales are equal for the transverse axes simultaneously. The small differences between the transverse scale error estimates (see the lower plots in

Figure 7) for each pair of estimates therefore do not indicate any overall disagreement with the MWD-error model. (This was also the case for insignificant scale error terms.) Axial scale errors. The axial scale was declared significantly different from zero for about 15 survey sections when the reference components were treated as measurements. These estimates were mostly positive and could be several percent in magnitude. Note that in the presence of the axial bias (which is usually more significant and therefore chosen in the expense of the axial scale) the true axial scale must be extraordinary large to be detected. This condition is strongly dependent on the inclination and azimuth variation. Since the estimated scales also mirror errors in the geomagnetic reference, it is not possible to conclude whether the model-misspecifications are due to sensor imperfections or errors in the geomagnetic components. The mismatch may also be due other underlying effects which cannot be modelled properly by the bias and scale factor errors only. Errors in the Earth's geomagnetic components. For the above mentioned surveys (where the axial scale became significant), the axial scale was excluded and the magnetic field strength B and dip angle θ were estimated. The result of this was that the magnetic field strength B differed significantly from zero for all these survey sections (often strongly significant), while the dip angle was significant only for a few. Some basic statistical parameters are given in Table 4. Since B and θ are closely related parameters, one should expect that if one of them is of poor quality, the other one will also be of poor quality. When considering this argumentation, the cause of the above disagreement is not the geomagnetic reference.

Parameter Mean Std.dev. Max. Min. B 655 nT 390 nT 1616 nT 215 nT

θ 0.13 deg. 0.13 deg. 0.43 deg 0.01 deg.

Table 4: Basic statistics of the absolute values of the difference between the a priori and estimated magnetic field strength and dip angle.

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THE FREQUENCY OF GROSS ERRORS Because undetected outliers might bias the estimated parameters considerably, this part of the quality control process is of crucial importance for the Multi-station analysis. Methodology. The test for the detection of outliers in individual sensor readings yi is based on the one dimensional version of the hypothesis in Equation (2). This test is considered useful for magnetic surveying quality control purposes, see [Nyrnes et al. 2005]. To specify the measurement which is most likely corrupted, the following datasnooping procedure is applied; by expanding the design matrix in Equation (2) with the unit vector ci = [0…0 1 0…0]T, and letting i vary from 1 to n (n is the number of measurements), the test-statistic in Equation (3) corresponding to every single error estimate i∇ is evaluated. This procedure can be carried out in a more straightforward way using residual analysis, see [Koch 1999], [Teunissen 2000] and [Nyrnes et al. 2005] The lower limit for the significance level αi of the individual tests is adjusted according to αi = α/n for survey sections with less than 35 stations, in order to obtain a type I error probability α of approximately 0.05 for the test of H0 (i.e. no outliers in the measurements). For longer wellsections, a lower limit of αi = 0.0001 is used for the two-tailed test. It is important to notice that the relation αi = α/n assumes independent tests, which is hardly the case. Despite of these simplifications and assumptions, it must be considered that the goal of applying such tests is to remove errors which may cause major harm to the estimation. It would not have considerable effects on the overall result if the lower limit for αi were set to 0.001 instead of 0.0001, for example. Outliers in the magnetometer measurements. The results are presented in the Figure 8 and the Tables 5 and 6. The estimates of the axial magnetometer errors are presented by the histogram to the upper in Figure 8. It can be seen that the largest amount of estimates are smaller than 500 nT. Some errors were estimated to be even larger than 4000 nT in absolute value. The cross-axial errors are shown in the lower plots in Figure 8. The majority of estimated biases are smaller than 1000 nT. Compared with other sources of errors such as borehole misalignments and declination errors, many outliers will in practice have no significant influence on the

computed wellbore position. However, removing them is important to achieve an optimal estimation. The main trend was that the measurements in the beginning of the sections corresponded to the largest test-statistics. This trend applied irrespective of the bore hole dimension and inclination. However, they were several times insignificant and thus they were not classified as outliers. Since this usually involved a maximum of three stations, the errors are divided into groups depending on whether they were detected at the first three stations, or at other stations. A summary is given in Table 5. Under similar conditions as in this study, one must generally expect to reject about 2 % of the axial magnetometer readings in parts below the first few stations, and about 1 % of the cross-axial readings.

Outliers detected in the axial magnetometer measurements.

Estimates (nT)

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000

Freq

uenc

y

02468

10121416

Outliers detected in the x-axial magnetometer measurements (nT).

Estimates (nT)

-750 0 750 1500 2250

Freq

uenc

y

0

2

4

6

8

10

Outliers detected in the y-axial magnetometer measurements (nT).

Estimates (nT)

-2000 -1000 0 1000 2000

Freq

uenc

y

0

2

4

6

8

10

Figure 8: Magnetometer measurements classified as outliers. Upper plot: Outliers detected in the axial measurements. The majority is smaller than 500 nT. Lower plots: Outliers detected in the cross-axial magnetometer readings. If more than one measurement at the same station is rejected, the ability to pinpoint the corrupted one out of the remaining two is almost impossible. It may also be that both the remaining measurements are corrupted. Basic information about the excluded stations is given in Table 6. It can be seen that almost all of the excluded stations belong to the five categories "St. no. 1" to "St. no. 5". The values in Tables 5 and 6 clearly indicate that the magnetic surveys performed at the first few stations in general are of the poorest quality. As an example,

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almost 40 % of all stations number 1 are corrupted by outliers, either in one or at least two measurements. Magnetic interference from the existing casing can explain the high frequency of large errors in the beginning of the wellsections. The errors may also be related to sensor failure, magnetic disturbances from the Earth's crust, human errors and sudden fluctuations in the Earth's magnetic field intensity.

Magnetometers Sensor St. 1 St. 2 St. 3 Elsewhere

bx 0 3 1 17 (0.9%) by 1 2 2 25 (1.3%) bz 13 (20%) 8 7 34 (1.8%)

Table 5: Total number of outliers in single sensor readings detected at the 3 first stations and at stations elsewhere in the well. The total number of measurements is about 1900.

Magnetometers St. 1 St. 2 St. 3

12 (18%) 9 (14%) 8 (12%) St. 4 St. 5 Elsewhere

5 (8%) 5 (8%) 5

Table 6: The number of excluded stations. If at least two magnetometer measurements turned out to be corrupted, the whole station was excluded.

Accelerometers Sensor Number of biases

gx 15 (0.8%) gy 8 (0.4%) gz 34 (1.7%)

Table 7: Outliers in single accelerometer readings out of the total number of measurements (approximately 2000) and percentage share. The axial accelerometer is more exposed to outliers than the cross-axial ones. Outliers in the accelerometer measurements. The ability to detect outliers in the accelerometer readings is poorer than for the magnetometer readings. In general, the errors have to be even larger to be detected. In addition, it may sometimes be more difficult to pinpoint the measurement which actually is corrupted. In directions close to the east-west the corrupted measurement cannot be pinpointed, since the residuals ê become totally correlated. In this case the contribution from the magnetometer measurements is given only in

terms of higher redundancy, thus increasing the power of the statistical tests. This is considered helpful when the actual wellbore provides low redundancy. For a few wellsections, several stations were rejected (totally 15), because more than one accelerometer measurement were most likely corrupted. A rather large number of outliers was also detected in single accelerometer measurements. The estimated outliers are shown in Figure 9. The majority of biases range between approximately ±5 Gal for all three axes. Table 7 shows that the z-axial errors are about two to three times more frequent than the transverse ones. One possible explanation is the movements of floating rigs, causing the z-axis sensor to become more restless than the transverse ones.

Outliers detected in the transverseacc. measurements

Estimated biases (Gal)

-15 -12 -9 -6 -3 0 3 6 9

Freq

uenc

y

0

2

4

6

8

10

12

14

Outliers detected in the z-axial acc. measurements

Estimated biases (Gal)-12 -9 -6 -3 0 3 6 9

Freq

uenc

y

0

2

4

6

8

10

12

14

Outliers detected in the transverseand z-axial acc. measurements

Estimated biases (Gal)-15 -10 -5 0 5 10

Freq

uenc

y

02468

10121416

Figure 9: Errors (estimated biases) classified as outliers. Unit: Gal. Upper plots: Outliers detected in the cross-axial and axial accelerometer measurements. Lower plot: All outliers lumped together. Most of the errors range between −5 Gal and 5 Gal for all three axis. Comments on the results. Despite of the low significance level used (about 0.1 ‰) the large number of detected outliers clearly indicates that the tail of the real distribution of the accelerometer and magnetometer reading errors deviates significantly from the assumed normal distribution. Removing them is therefore important in order to avoid unwanted biases in the estimates.

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RANDOM NOISE In this context, noise is expressed as the estimated uncertainty of the errors e of the accelerometer and magnetometer readings. This quantity is of crucial importance for the analyses. In general, a high noise level results in low power for the statistical tests of significance. The estimated in situ uncertainty

iyσ of a particular sensor reading yi is given by:

1iy pˆˆ

i

−σ=σ (6)

where pi is the weight of the measurement and 2σ the estimate of the variance σ2 of unit weight. See Appendix C for details. The dominant error source in magnetometer measurements is usually irregular variations in the Earth's magnetic field. Such time and location dependent variations are due to the complicated interactions between the magnetic field and Solar winds [Torkildsen et al. 1997]. However, irregular errors related to the performance of the survey tool may also have significant effects when solar activity is low. The gravity field can be considered as a stable reference over time. However, the accelerometer measurements may be affected by vibrations. This is expected to be the dominant error source and will usually mask the effects of random sensor errors completely.

Estimated uncertainties of accelerometer reading errors (Gal)

Standard deviation (Gal)

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Freq

uenc

y

02468

10121416182022

Estimated uncertainties of magnetometer reading errors (nT)

Standard deviation (nT)

0 20 40 60 80 100120140160180200

Freq

uenc

y

02468

10121416182022

Figure 10: The estimated noise standard deviations in the accelerometer and magnetometer measurements. Comments on the results. The estimated standard deviations of the errors in the accelerometer readings are given in the left plot in Figure 10. Nearly 55 % of the estimates have their values within the interval: 0.15 Gal - 0.25 Gal. Moreover, about 20 % of the estimated standard deviations are larger than 0.25 Gal, and almost 15 % are even larger than 0.4 Gal. The two most extreme noise levels are approximately 17 Gal (not

included in the histogram) and 0.027 Gal, both representing relatively steep 17.5 inch sections.

Figure 11: Upper plot: Standard deviations (linear interpolated between 69.66, 62.07 and 60.13 degrees latitude) are extrapolated assuming an 11 year periodic solar activity. The standard deviations are based on the Geomagnetic Reference Report [Torkildsen et al. 1997]. Lower plot: Estimated noise in magnetic survey data vs. latitude and years. The histogram to the right in Figure 10 shows that the estimated standard deviations of the magnetometer measurements are lower than 100 nT for almost 80 % of the survey sections, while about 60 % are lower than 60 nT. One is even lower than 20 nT. The estimates are plotted against latitude and years, see the lower plot in Figure 11 and also Figure 12. The noise levels are highest for northern areas and lowest for southern areas. The same trend can also be seen from the contour plot in the upper part of Figure 11, where the noise levels for the years 1991-1996 are based on the time and location dependent variations (noise) of the magnetic field observed at monitor stations in the North Sea and Norwegian Ocean area [Torkildsen et al. 1997]. The plot also shows the future noise predicted under the assumption of a sun activity cycle of 11 years. A comparison between the upper and lower plots in Figure 11 show some similar trends, especially for the

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latitude dependent noise. The results indicate that the estimated weight relations between the accelerometer and magnetometer readings are reasonable. The results also confirm that the major effects of outliers and systematic errors are removed from the measurements. It is important to notice that the level of irregular variations of the magnetic field intensity in a shorter period of time may differ considerably from the average over a longer period of time. If the estimated noise level turns out to be unexpectedly high, it does not necessarily mean that the functional model (Equation (1)) is misspecified. The reason might be that the surveys are performed in a period with higher solar activity than normal, for example in the case of magnetic storms.

Noise standard deviation (nT)versus years.

Year

1998 1999 2000 2001 2002 2003 2004 2005 2006

Estim

ated

stan

dard

dev

iatio

n (n

T)

0

20

40

60

80

100

120

140

160

180

Estimated std.dev. of magnetometermeasurements (nT).

Noise standard deviation (nT)versus latitude.

Latitude

56 58 60 62 64 66 68

Estim

ated

stan

dard

dev

iatio

n (n

T)

0

20

40

60

80

100

120

140

160

180

Estimated std.dev. of magnetometermeasurements (nT).

Figure 12: Left hand plot: The estimated survey uncertainty (noise) increases for greater latitudes. Right hand plot: The noise is lowest for the most recent surveys.

SUMMARY AND CONCLUSIONS The error properties of magnetic directional surveys from more than 60 wellbore sections have been investigated using multi-station estimation techniques.

The most important results are summarized in Table 8. The ISCWSA error term values. First, the properties of systematic error terms were investigated. The International Steering Committee on Wellbore Survey Accuracy (ISCWSA) has developed error models for magnetic directional surveys (MWD). The results showed that the cross-axial accelerometer bias errors were in accordance with the error model, while the magnetometer bias errors tended to be somewhat lower than modelled. Because the estimation of the axial accelerometer bias and scale error terms involved several uncertain factors, they were not considered. For the cross-axial accelerometer scale factor errors the estimates showed a tendency to larger values than modelled. However,

when taken into consideration the relatively high chance for the a priori local gravity to be of poor quality, the estimated error values do not show an overall disagreement with the MWD-error model. For the surveys which had been corrected for axial magnetic interference, the estimated magnetometer bias errors can be considered to be fairly in accordance with the MWD-error model. The estimates of the cross-axial magnetometer scale errors were up to a few percent in magnitude for several surveys. The errors were insignificantly different from zero for about 60 % of the surveys. It was pointed out that due to the high noise level in the measurements, it will usually not be possible to detect errors smaller than the corresponding MWD-error model value (0.16 %). Therefore, any agreement or disagreement with the MWD-error model could not be proved. Since the transverse error estimates were similar and negative (indication of damping), the magnetic properties of the drilling fluid was most likely the reason. Sometimes the axial scale errors turned out to be clearly significant. It was shown that this also could be attributed to errors in the Earth's magnetic reference components, and it could therefore not be proved whether sensor imperfections were the cause. Gross errors. It was demonstrated how multi-station estimation techniques can be used to detect gross errors in single sensor readings. Removing these errors is necessary to obtain an optimal estimation. The first few stations were most critical for the magnetometers, probably due to magnetic interference from existing casing. The worst case was for the first station of the survey section. About 40 % of them were affected by outliers. However, it was often sufficient to reject the z-axis measurement only. In parts below the first few stations, almost 2 % of the z-axis magnetometer measurements were rejected, and about 1 % of the cross-axial readings. For the accelerometers, up to 2 % of the z-axial measurements were rejected, and about 1 percent of the cross-axial readings. The large number of detected outliers despite of the low significance level used (about 0.1 ‰) indicated that the normality assumption of the measurement errors is not reasonable. Random errors. Finally, the properties of the random errors ("noise") in the surveys were evaluated. The estimated noise standard deviations in the accelerometer readings ranged between 0.15 Gal and 0.25 Gal for about 50 % of the examined sections. For an equivalent number of magnetometer readings the standard deviations ranged between 40 nT and 60 nT.

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The surveys performed in northern areas could be several times noisier than surveys performed 5 degrees farther south. The estimated noise levels were in accordance with the expected noise levels of the magnetic field. These agreements indicate that the weight relations between the accelerometers and magnetometers are reasonable, and that the major effects of outliers and systematic errors are removed from the measurements.

Accelerometers

Sensor Error term ISCWSA RMS Outliers

rand. - 0.34 Gal gx bias 0.39 Gal 0.47 / 0.37 Gal 0.8 % scale 0.05 % 0.084 %

rand. - 0.34 Gal

gy bias 0.39 Gal 0.46 / 0.38 Gal 0.4 % scale 0.05 % 0.096 %

rand. - 0.34 Gal

gz bias 0.39 Gal - 1.7 % scale 0.05 % -

Magnetometers rand. - 73 nT

bx bias 70 nT 95 / 64 nT 0.9 % scale 0.16 % - 1.06 / 0.65 %

rand. - 73 nT

by bias 70 nT 72 / 47 nT 1.3 % scale 0.16 % 1.01 / 0.62 %

rand. - 73 nT

bz bias ≈ 165 nT 594 nT 1.8 % scale 0.16 % -

Table 8: Third column: Error values according to the MWD-error model developed by ISCWSA. Fourth column: RMS-values based on the estimates corresponding to significant parameters. RMS-values behind virgule include zero values for insignificant parameters. The RMS for the axial magnetometer bias is representative for surveys which are corrected for axial magnetic interference. Fifth column: Number of single sensor readings rejected (percent) in parts below the first three stations.

The quality of the Earth’s reference components. The results of the analyses showed that the gravity values often provide insufficiency quality for multi station estimation purposes.

The results showed that the quality of the input Earth's magnetic field intensity sometimes may be worse than expected. Except from this, nothing indicated that the overall quality of the magnetic field intensity and dip angle were unacceptable.

ACKNOWLEDGEMENTS The authors acknowledge Statoil ASA for permission to publish this paper and for providing the survey data.

APPENDIX A: COORDINATE SYSTEMS

(Plumb line)

ybbx

bz

y

xHigh sidedirection

Horizontal plane

Inclination

High sidetoolface

V

xggy

zg z

directionHigh side

z

Figure 13: Figurative definition of the instrument xyz-coordinate system, high side toolface and inclination. The x- and y-axes are often referred to as the cross-axial direction, transverse to the along-hole (z-axial/axial) direction.

N

E

Magneticnorth

Horizontalwellbore

A

A

m direction

Bfield vector)(Earth's magnetic

Horizontal plane

V(Plumb line)

Figure 14: Figurative definitions of the azimuth A, magnetic azimuth Am, magnetic declination δ and the magnetic dip angle Θ. At a particular point, δ is defined as the angular difference between the horizontal component of the Earth's magnetic field vector and the true north. It is by definition positive when magnetic north lies east of true north, and negative when magnetic north lies west of true north.

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APPENDIX B: THE GAUSS-NEWTON METHOD Again, we consider the linear regression model introduced in Equation (1):

[ ] β

Z Xeyy ⎥⎦

⎤⎢⎣

⎡∇

=−=dd

)(E (7)

with E(e) = 0, Cov(e) = Cov(y) = Σee = σ2Qee. These equations are often referred to as observation equations or just error equations. In our case the functional relationship between E(y) and [β ∇]T is nonlinear, so instead of Equation (8) we get the inconsistent system of nonlinear equations:

[ ] ) ( T∇=− βhey (8) An approach to solve this nonlinear least squares problem is that of Gauss-Newton. Using the first two terms of a Taylor expansion of Equation (9) to form a linear approximation about initial values [ ]T00 ∇ β gives:

[ ] ⎥⎦

⎤⎢⎣

⎡∇

+∇=

=∇+∇+≈−

dd

),(

) ,d

00

00

βZ Xβh

dβh(βey (9)

where h is a vector of nonlinear functions of the unknowns, [ ]Z X is a matrix of partial derivatives with respect to the elements in [ ]∇ β .

The least squares estimator [ ]Tˆdˆd ∇ β of the unknown

corrections [ ]Tdd ∇ β is obtained by solving the Equations (10) iteratively subject to the constraint

eQe 1ee

T − = minimum. The unknown corrections are estimated by:

PfXPXXβ T1T )(ˆd −= (10) The i'th element in f can be considered as the difference between the actual measurement and the approximate measurement:

[ ] 0T

r21u21iii ),..., ,...,,(hyf ∇∇∇βββ−= (11) The estimator β is approximated by the last update of:

k0,..., i ,ˆdˆˆii1i ∈+=+ βββ (12)

where k is the number of iterations needed to make the elements in βd smaller than some preset amount close to zero. See [Koch 1999] for more details.

APPENDIX C: THE PROPERTIES OF LEAST SQUARES The properties of the least squares estimators are in the following given in accordance with the theory of linear algebra, see [Koch 1999]. Theoretically, these properties apply only approximately for nonlinear least squares. Suppose in the following that β represents all fixed parameters (angular components and systematic errors). It can be shown that if )σ,0(N~ 12 −Pe , the least squares estimators are normally distributed:

))(,(N~ˆ 1T2 −σ PXXββ (13)

))(,(N~)(E T1T2 XPXXXXβy −σ (14)

))((,0(N~ˆ T1T12 XPXXXPe −− −σ (15) The weighted residual sum of squares Ω is given by:

ePe ˆˆT=Ω (16) which is chi-squared distributed with n-u degrees of freedom, 2

)un(2~ −χσΩ with mean .un)(E −=Ω From

this it follows that the unbiased estimator 2σ of the variance 2σ of unit weight can be computed from:

ununˆˆˆ

T2

−Ω

=−

=σePe (17)

and distributed as:

)un(~ˆ)un( 2

2

2−χ

σσ− (18)

The residuals e are for nonlinear least squares given by

)(Eˆ yye −= , where the elements of )(E y are given by the last update of the approximate measurements 0h (defined in Equation (11), Appendix B). In the case when 2σ is unknown, the covariance matrix ββ ˆˆC of

β is estimated by:

1T2ˆˆ

2ˆˆ )(ˆˆˆ −

ββββ σ=σ= PXXQC (19)

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where Q is the matrix of cofactors. See [Koch 1999] for further details. The application of the least squares criterion requires no prior distribution about the measurement errors to be specified. However, in the case of normally distributed observation errors, the method of least squares and the maximum likelihood method lead to identical unbiased estimators for the unknown parameters. However, the least squares criterion applied to normally distributed errors also provide unbiased and minimum variance properties for the estimate 2σ of the variance 2σ of unit weight.

APPENDIX D: GRAVITY REFERENCE FOR DIRECTIONAL SURVEYING Using precise gravity values is important when

• comparing G derived from 3-component accelerometer readings with the nominal value.

• using 2-accelerometer systems for determination of inclination.

This appendix presents a simplified and generalized approach to calculate G. The formula precision is better than 0.0005 m/s2 (50 mGal), and is assumed to be precise enough for the quality control of directional surveying services. However, the service companies should derive G-reference values in a more reliable way for calibration purposes. Formulas. The main trend is according to the Geodetic Reference System 1980. The original formula is truncated and given a best fit globally (within 3 mgal precision). Additionally the Free-air, Bouguer and regional Isostatic corrections are applied. General formula:

( )TVDρ084.0)H(TVD309.0

sin5186978030G

0

2

⋅⋅−−⋅+ϕ⋅+= (20)

where G is the predicted gravity in mgal (1mgal = 10-5m/s2), H0 is the height above MSL at installation in metres, TVD is the vertical depth in metres measured from installation, ϕ is the latitude and ρ is the bulk density in g/cm3 from installation to the actual TVD. The missing Anomaly, Topographic and local Isostatic corrections may cause significant errors (>50 mgals) in mountainous areas. The above formula can be simplified further for:

• offshore sites by using a bulk density of 2.0 g/cm3, and approximating installation height to 0 m.

• onshore sites by using a bulk density of 2.5 g/cm3 .

Offshore formula:

( ) TVD14.0sin5186978030G 2 ⋅+ϕ⋅+= (21) Onshore formula:

( )0

2

H31.0TVD10.0sin5186978030G

⋅−⋅+ϕ⋅+= (22)

See [Moritz 2000]. See also different textbooks, for example [Heiskanen, Moritz 1993].

REFERENCES Brooks AG, Gurden PA, Noy KA (1998) Practical

Application of a Multiple-Survey Magnetic Correction Algorithm. Paper SPE 49060, 1998 SPE Annual Technical Conference and Exhibition, New Orleans, 27 September – 30 October 1998

Ekseth R (1998) Uncertainties in connection with the determination of wellbore positions. ISBN 82-471-0218-8, Doctoral thesis 1998, Norwegian University of Science and Technology, 1998:24 IPT-rapport

Heiskanen WA, Moritz H (1993) Physical geodesy. Technical University Graz, 1993

Koch KR (1999) Parameter Estimation and Hypothesis Testing in Linear Models. ISBN 3-540-65257-4, Springer-Verlag 1999

Miller AJ (2002) Subset Selection in Regression. 2nd edition, ISBN 1584881712, Chapman & Hall/CRC, 2002

Moritz H (2000) Geodetic Reference System 1980 http://www. gfy.ku.dk/~iag/HB2000/part4/grs80_corr.htm

Nyrnes E, Torkildsen T, Nahavandchi H (2005) Detection of Gross Errors in Wellbore Directional Surveying with Emphasis on Reliability Analyses. Accepted for publication in Kart og Plan vol. 2 2005

Teunissen PJG (2000) Testing theory: an introduction. ISBN 90-407-1975-6, Delft University Press 2000

Torkildsen T, Sveen RH, Bang J (1997) Time Dependent Variations of Declination. Geomagnetic Reference report No. 1, IKU Petroleum Research 1997

Torkildsen T, Edwardsen I, Fjogstad A, Saasen A, Amundsen PA, Omland TH (2004) Drilling Fluid affects MWD Magnetic Azimuth and Wellbore Position. Paper SPE 87169, 2004, IADC/SPE Drilling Conference Dallas, Texas, 2-4 March 2004

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Williamson HS (2000) Accuracy Prediction for Directional Measurement While Drilling. Paper SPE 67616, SPE Drilling and Completion 15 (4), December 2000

Wilson H, Brooks AG (2001) Wellbore Position Errors Caused by Drilling Fluid Contamination. Paper SPE 71400, 2001 SPE Annual Technical Conference and Exhibition, New Orleans, 30 September – 3 October 2001

ABOUT THE AUTHORS Erik Nyrnes is a research fellow at the Norwegian University of Science and Technology (NTNU). He holds a master’s degree in surveying from the same university. Torgeir Torkildsen is a specialist in directional surveying at Statoil ASA Norway. He holds a doctoral degree in geodesy from NTNU. Hossein Nahavandchi is an associate professor in geodesy at NTNU. He holds a doctoral degree in geodesy from the Royal Institute of Technology, Sweden.